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<h2>SL Paper 1</h2><div class="specification">
<p><strong>In this question, all lengths are in metres and time is in seconds.</strong></p>
<p>Consider two particles, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>2</mn></msub></math>, which start to move at the same time.</p>
<p>Particle <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>1</mn></msub></math> moves in a straight line such that its displacement from a fixed-point is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mn>10</mn><mo>-</mo><mfrac><mn>7</mn><mn>4</mn></mfrac><msup><mi>t</mi><mn>2</mn></msup></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 an expression for the velocity of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>1</mn></msub></math> 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">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Particle <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>2</mn></msub></math> also moves in a straight line. The position of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>2</mn></msub></math> is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>6</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>t</mi><mfenced><mtable><mtr><mtd><mn>4</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>3</mn></mtd></mtr></mtable></mfenced></math>.</p>
<p>The speed of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>1</mn></msub></math> is greater than the speed of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>2</mn></msub></math> when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>></mo><mi>q</mi></math>.</p>
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>recognizing velocity is derivative of displacement <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>=</mo><mfrac><mrow><mtext>d</mtext><mi>s</mi></mrow><mrow><mtext>d</mtext><mi>t</mi></mrow></mfrac><mo> </mo><mo>,</mo><mo> </mo><mfrac><mtext>d</mtext><mrow><mtext>d</mtext><mi>t</mi></mrow></mfrac><mfenced><mrow><mn>10</mn><mo>-</mo><mfrac><mn>7</mn><mn>4</mn></mfrac><msup><mi>t</mi><mn>2</mn></msup></mrow></mfenced></math></p>
<p>velocity<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mfrac><mn>14</mn><mn>4</mn></mfrac><mi>t</mi><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mo>-</mo><mfrac><mn>7</mn><mn>2</mn></mfrac><mi>t</mi></mrow></mfenced></math> <em><strong>A1 N2</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>valid approach to find speed of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>2</mn></msub></math> <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mfenced><mtable><mtr><mtd><mn>4</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>3</mn></mtd></mtr></mtable></mfenced></mfenced><mo> </mo><mo>,</mo><mo> </mo><msqrt><msup><mn>4</mn><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mo>-</mo><mn>3</mn></mrow></mfenced><mn>2</mn></msup></msqrt></math> , velocity<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msqrt><msup><mn>4</mn><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mo>-</mo><mn>3</mn></mrow></mfenced><mn>2</mn></msup></msqrt></math></p>
<p>correct speed <em><strong>(A1)</strong></em></p>
<p><em>eg </em><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo> </mo><msup><mtext>m s</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></p>
<p>recognizing relationship between speed and velocity (may be seen in inequality/equation) <em><strong>R1</strong></em></p>
<p><em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mrow><mo>-</mo><mfrac><mn>7</mn><mn>2</mn></mfrac><mi>t</mi></mrow></mfenced></math> , speed = | velocity | , graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>1</mn></msub></math> speed , <img src="data:image/png;base64,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"> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>1</mn></msub></math> speed <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>7</mn><mn>2</mn></mfrac><mi>t</mi><mo> </mo><mo>,</mo><mo> </mo><msub><mi>P</mi><mn>2</mn></msub></math> velocity <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mn>5</mn></math></p>
<p>correct inequality or equation that compares speed or velocity (accept any variable for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math>) <em><strong>A1</strong></em></p>
<p><em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mrow><mo>-</mo><mfrac><mn>7</mn><mn>2</mn></mfrac><mi>t</mi></mrow></mfenced><mo>></mo><mn>5</mn><mo> </mo><mo>,</mo><mo> </mo><mo>-</mo><mfrac><mn>7</mn><mn>2</mn></mfrac><mi>q</mi><mo><</mo><mo>-</mo><mn>5</mn><mo> </mo><mo>,</mo><mo> </mo><mfrac><mn>7</mn><mn>2</mn></mfrac><mi>q</mi><mo>></mo><mn>5</mn><mo> </mo><mo>,</mo><mo> </mo><mfrac><mn>7</mn><mn>2</mn></mfrac><mi>q</mi><mo>=</mo><mn>5</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>=</mo><mfrac><mn>10</mn><mn>7</mn></mfrac></math> (seconds) (accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>></mo><mfrac><mn>10</mn><mn>7</mn></mfrac></math> , do not accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mfrac><mn>10</mn><mn>7</mn></mfrac></math>) <em><strong>A1 N2</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Do not award the last two <em><strong>A1</strong></em> marks without the <em><strong>R1</strong></em>.</p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Sieun hits golf balls into the air. Each time she hits a ball she records <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math>, the angle at which the ball is launched into the air, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi></math>, the horizontal distance, in metres, which the ball travels from the point of contact to the first time it lands. The diagram below represents this information.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p>Sieun analyses her results and concludes:</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>l</mi></mrow><mrow><mo>d</mo><mi>θ</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>2</mn><mi>θ</mi><mo>+</mo><mn>9</mn><mo>,</mo><mo> </mo><mo> </mo><mn>35</mn><mo>°</mo><mo>≤</mo><mi>θ</mi><mo>≤</mo><mn>75</mn><mo>°</mo></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine whether the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi></math> against <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math> is increasing or decreasing at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mo>=</mo><mn>50</mn><mo>°</mo></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>Sieun observes that when the angle is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>40</mn><mo>°</mo></math>, the ball will travel a horizontal distance of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>205</mn><mo>.</mo><mn>5</mn><mo> </mo><mtext>m</mtext></math>.</p>
<p>Find an expression for the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi><mfenced><mi>θ</mi></mfenced></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi><mo>'</mo><mfenced><mn>50</mn></mfenced><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>2</mn><mo>×</mo><mn>50</mn><mo>+</mo><mn>9</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p>the curve is decreasing at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mo>=</mo><mn>50</mn><mo>°</mo></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> For the final <em><strong>A1</strong></em>, follow through within this question part for their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi><mo>'</mo><mfenced><mn>50</mn></mfenced></math> value. Award <em><strong>A0</strong></em> for an answer of "decreasing" with no work shown.</p>
<p><em><strong><br>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognition of need to integrate (e.g. reverse power rule or integral symbol or integrating at least one term correctly) <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi><mfenced><mi>θ</mi></mfenced><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>θ</mi><mn>2</mn></msup><mo>+</mo><mn>9</mn><mi>θ</mi><mo> </mo><mfenced><mrow><mo>+</mo><mi>c</mi></mrow></mfenced></math> <em><strong>A1A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>205</mn><mo>.</mo><mn>5</mn><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>1</mn><mo>×</mo><msup><mfenced><mn>40</mn></mfenced><mn>2</mn></msup><mo>+</mo><mn>9</mn><mo>×</mo><mfenced><mn>40</mn></mfenced><mo>+</mo><mi>c</mi></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>M1</strong></em> for correct substitution of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mo>=</mo><mn>40</mn><mo>°</mo></math> <strong>and</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi><mo>=</mo><mn>205</mn><mo>.</mo><mn>5</mn></math>. A constant of integration must be seen (can be implied by a correct answer).</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>5</mn><mo>.</mo><mn>5</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>l</mi><mfenced><mi>θ</mi></mfenced><mo>=</mo></mrow></mfenced><mo>-</mo><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>θ</mi><mn>2</mn></msup><mo>+</mo><mn>9</mn><mi>θ</mi><mo>+</mo><mn>5</mn><mo>.</mo><mn>5</mn></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Accept any variable in the working, but for the final <em><strong>A1</strong></em>, the variable <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math> must be used in the expression.</p>
<p><em><strong><br>[5 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The following diagram shows part of the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mi>k</mi><mi>x</mi></mfrac></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mn>0</mn><mo>,</mo><mo> </mo><mi>k</mi><mo>></mo><mn>0</mn></math>.</p>
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>p</mi><mo>,</mo><mo> </mo><mfrac><mi>k</mi><mi>p</mi></mfrac></mrow></mfenced></math> be any point on the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math>. Line <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>1</mn></msub></math> is the tangent to the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>.</p>
<p style="text-align: center;"><img 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"></p>
</div>
<div class="specification">
<p>Line <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>1</mn></msub></math> intersects the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis at point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mfenced><mrow><mn>2</mn><mi>p</mi><mo>,</mo><mo> </mo><mn>0</mn></mrow></mfenced></math> and the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis at point B.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>p</mi></mfenced></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the equation of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>1</mn></msub></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mi>x</mi><mo>+</mo><msup><mi>p</mi><mn>2</mn></msup><mi>y</mi><mo>-</mo><mn>2</mn><mi>p</mi><mi>k</mi><mo>=</mo><mn>0</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 area of triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>AOB</mtext></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>.</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 graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is translated by <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>4</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd></mtr></mtable></mfenced></math> to give the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math>.<br>In the following diagram:</p>
<ul>
<li>point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Q</mtext></math> lies on the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math>
</li>
<li>points <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>C</mtext></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>D</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>E</mtext></math> lie on the vertical asymptote of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math>
</li>
<li>points <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>D</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>F</mtext></math> lie on the horizontal asymptote of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math>
</li>
<li>point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>G</mtext></math> lies on the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>FG</mtext></math> is parallel to <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>DC</mtext></math>.</li>
</ul>
<p>Line <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>2</mn></msub></math> is the tangent to the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Q</mtext></math>, and passes through <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>E</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>F</mtext></math>.</p>
<p><img src="data:image/png;base64,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"></p>
<p>Given that triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>EDF</mtext></math> and rectangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>CDFG</mtext></math> have equal areas, find the gradient of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>2</mn></msub></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>-</mo><mi>k</mi><msup><mi>x</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></math> <em><strong> (A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>p</mi></mfenced><mo>=</mo><mo>-</mo><mi>k</mi><msup><mi>p</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mo>-</mo><mfrac><mi>k</mi><msup><mi>p</mi><mn>2</mn></msup></mfrac></mrow></mfenced></math> <em><strong> A1 N2</strong></em></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>attempt to use point and gradient to find equation of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>1</mn></msub></math> <em><strong>M1</strong></em></p>
<p><em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>-</mo><mfrac><mi>k</mi><mi>p</mi></mfrac><mo>=</mo><mo>-</mo><mi>k</mi><msup><mi>p</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>p</mi></mrow></mfenced><mo>,</mo><mo> </mo><mo> </mo><mfrac><mi>k</mi><mi>p</mi></mfrac><mo>=</mo><mo>-</mo><mfrac><mi>k</mi><msup><mi>p</mi><mn>2</mn></msup></mfrac><mfenced><mi>p</mi></mfenced><mo>+</mo><mi>b</mi></math></p>
<p>correct working leading to answer <em><strong> A1</strong></em></p>
<p><em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mi>y</mi><mo>-</mo><mi>k</mi><mi>p</mi><mo>=</mo><mo>-</mo><mi>k</mi><mi>x</mi><mo>+</mo><mi>k</mi><mi>p</mi><mo>,</mo><mo> </mo><mo> </mo><mi>y</mi><mo>-</mo><mfrac><mi>k</mi><mi>p</mi></mfrac><mo>=</mo><mo>-</mo><mfrac><mi>k</mi><msup><mi>p</mi><mn>2</mn></msup></mfrac><mi>x</mi><mo>+</mo><mfrac><mi>k</mi><mi>p</mi></mfrac><mo>,</mo><mo> </mo><mo> </mo><mi>y</mi><mo>=</mo><mo>-</mo><mfrac><mi>k</mi><msup><mi>p</mi><mn>2</mn></msup></mfrac><mi>x</mi><mo>+</mo><mfrac><mrow><mn>2</mn><mi>k</mi></mrow><mi>p</mi></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mi>x</mi><mo>+</mo><msup><mi>p</mi><mn>2</mn></msup><mi>y</mi><mo>-</mo><mn>2</mn><mi>p</mi><mi>k</mi><mo>=</mo><mn>0</mn></math> <em><strong> AG N0</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1 – area of a triangle</strong></p>
<p>recognizing <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> <em><strong>(M1)</strong></em></p>
<p>correct working to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-coordinate of null<em><strong> (A1)</strong></em></p>
<p><em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mi>y</mi><mo>-</mo><mn>2</mn><mi>p</mi><mi>k</mi><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-coordinate of null at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mrow><mn>2</mn><mi>k</mi></mrow><mi>p</mi></mfrac></math> (may be seen in area formula) <em><strong> A1</strong></em></p>
<p>correct substitution to find area of triangle<em><strong> (A1)</strong></em></p>
<p><em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mrow><mn>2</mn><mi>p</mi></mrow></mfenced><mfenced><mfrac><mrow><mn>2</mn><mi>k</mi></mrow><mi>p</mi></mfrac></mfenced><mo>,</mo><mo> </mo><mo> </mo><mi>p</mi><mo>×</mo><mfenced><mfrac><mrow><mn>2</mn><mi>k</mi></mrow><mi>p</mi></mfrac></mfenced></math></p>
<p>area of triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>AOB</mtext><mo>=</mo><mn>2</mn><mi>k</mi></math> <em><strong> A1 N3</strong></em></p>
<p> </p>
<p><strong>METHOD 2 – integration</strong></p>
<p>recognizing to integrate <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>1</mn></msub></math> between <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>p</mi></math> <em><strong>(M1)</strong></em></p>
<p><em>eg </em> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mrow><mn>2</mn><mi>p</mi></mrow></msubsup><msub><mi>L</mi><mrow><mn>1</mn><mo> </mo></mrow></msub><mo>d</mo><mi>x</mi><mo> </mo><mo>,</mo><mo> </mo><msubsup><mo>∫</mo><mn>0</mn><mrow><mn>2</mn><mi>p</mi></mrow></msubsup><mo>-</mo><mfrac><mi>k</mi><msup><mi>p</mi><mn>2</mn></msup></mfrac><mi>x</mi><mo>+</mo><mfrac><mrow><mn>2</mn><mi>k</mi></mrow><mi>p</mi></mfrac></math></p>
<p>correct integration of <strong>both</strong> terms <em><strong> A1</strong></em></p>
<p><em>eg </em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mrow><mi>k</mi><msup><mi>x</mi><mn>2</mn></msup></mrow><mrow><mn>2</mn><msup><mi>p</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn><mi>k</mi><mi>x</mi></mrow><mi>p</mi></mfrac><mo> </mo><mo>,</mo><mo> </mo><mo>-</mo><mfrac><mi>k</mi><mrow><mn>2</mn><msup><mi>p</mi><mn>2</mn></msup></mrow></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mrow><mn>2</mn><mi>k</mi></mrow><mi>p</mi></mfrac><mi>x</mi><mo>+</mo><mi>c</mi><mo> </mo><mo>,</mo><mo> </mo><msubsup><mfenced open="[" close="]"><mrow><mo>-</mo><mfrac><mi>k</mi><mrow><mn>2</mn><msup><mi>p</mi><mn>2</mn></msup></mrow></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mrow><mn>2</mn><mi>k</mi></mrow><mi>p</mi></mfrac><mi>x</mi></mrow></mfenced><mn>0</mn><mrow><mn>2</mn><mi>p</mi></mrow></msubsup></math></p>
<p>substituting limits into <strong>their</strong> integrated function and subtracting (in either order) <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mrow><mi>k</mi><msup><mfenced><mrow><mn>2</mn><mi>p</mi></mrow></mfenced><mn>2</mn></msup></mrow><mrow><mn>2</mn><msup><mi>p</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn><mi>k</mi><mfenced><mrow><mn>2</mn><mi>p</mi></mrow></mfenced></mrow><mi>p</mi></mfrac><mo>-</mo><mfenced><mn>0</mn></mfenced><mo>,</mo><mo> </mo><mo>-</mo><mfrac><mrow><mn>4</mn><mi>k</mi><msup><mi>p</mi><mn>2</mn></msup></mrow><mrow><mn>2</mn><msup><mi>p</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>4</mn><mi>k</mi><mi>p</mi></mrow><mi>p</mi></mfrac></math></p>
<p>correct working<em><strong> (A1)</strong></em></p>
<p><em>eg </em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>4</mn><mi>k</mi></math></p>
<p>area of triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>AOB</mtext><mo>=</mo><mn>2</mn><mi>k</mi></math> <em><strong> A1 N3</strong></em></p>
<p> </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><strong>Note:</strong> In this question, the second <em><strong>M</strong></em> mark may be awarded independently of the other marks, so it is possible to award <em><strong>(M0)(A0)M1(A0)(A0)A0</strong></em>.</p>
<p> </p>
<p>recognizing use of transformation <em><strong>(M1)</strong></em></p>
<p><em>eg</em> area of triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>AOB</mtext></math> = area of triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>DEF</mtext></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mi>k</mi><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfrac><mo>+</mo><mn>3</mn><mo>,</mo></math> gradient of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>2</mn></msub><mo>=</mo></math> gradient of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>1</mn></msub></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>D</mtext><mfenced><mrow><mn>4</mn><mo>,</mo><mo> </mo><mn>3</mn></mrow></mfenced><mtext>, 2p+4, </mtext></math> one correct shift</p>
<p>correct working<em><strong> (A1)</strong></em></p>
<p><em>eg</em> area of triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>DEF</mtext><mo>=</mo><mn>2</mn><mi>k</mi><mo>,</mo><mo> </mo><mtext>CD</mtext><mo>=</mo><mn>3</mn><mo>,</mo><mo> </mo><mtext>DF</mtext><mo>=</mo><mn>2</mn><mi>p</mi><mo>,</mo><mo> </mo><mtext>CG</mtext><mo>=</mo><mn>2</mn><mi>p</mi><mo>,</mo><mo> </mo><mtext>E</mtext><mfenced><mrow><mn>4</mn><mo>,</mo><mo> </mo><mfrac><mrow><mn>2</mn><mi>k</mi></mrow><mi>p</mi></mfrac><mo>+</mo><mn>3</mn></mrow></mfenced><mo>,</mo><mo> </mo><mtext>F</mtext><mfenced><mrow><mn>2</mn><mi>p</mi><mo>+</mo><mn>4</mn><mo>,</mo><mo> </mo><mn>3</mn></mrow></mfenced><mo>,</mo><mo> </mo><mtext>Q</mtext><mfenced><mrow><mi>p</mi><mo>+</mo><mn>4</mn><mo>,</mo><mo> </mo><mfrac><mi>k</mi><mi>p</mi></mfrac><mo>+</mo><mn>3</mn></mrow></mfenced><mo>,</mo></math> </p>
<p>gradient of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>2</mn></msub><mo>=</mo><mo>-</mo><mfrac><mi>k</mi><msup><mi>p</mi><mn>2</mn></msup></mfrac><mo>,</mo><mo> </mo><mi>g</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>-</mo><mfrac><mi>k</mi><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup></mfrac><mo>,</mo></math> area of rectangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>CDFG</mtext><mo>=</mo><mn>2</mn><mi>k</mi></math></p>
<p>valid approach <em><strong>(M1)</strong></em></p>
<p><em>eg </em><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mtext>ED</mtext><mo>×</mo><mtext>DF</mtext></mrow><mn>2</mn></mfrac><mo>=</mo><mtext>CD</mtext><mo>×</mo><mtext>DF</mtext><mo>,</mo><mo> </mo><mn>2</mn><mi>p</mi><mo>·</mo><mn>3</mn><mo>=</mo><mn>2</mn><mi>k</mi><mo> </mo><mo>,</mo><mo> </mo><mtext>ED</mtext><mo>=</mo><mn>2</mn><mtext>CD</mtext><mo> </mo><mo>,</mo><mo> </mo><msubsup><mo>∫</mo><mn>4</mn><mrow><mn>2</mn><mi>p</mi><mo>+</mo><mn>4</mn></mrow></msubsup><msub><mi>L</mi><mn>2</mn></msub><mo> </mo><mtext>d</mtext><mi>x</mi><mo>=</mo><mn>4</mn><mi>k</mi></math></p>
<p>correct working<em> <strong>(A1)</strong></em></p>
<p><em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>ED</mtext><mo>=</mo><mn>6</mn><mo>,</mo><mo> </mo><mtext>E</mtext><mfenced><mrow><mn>4</mn><mo>,</mo><mo> </mo><mn>9</mn></mrow></mfenced><mo>,</mo><mo> </mo><mi>k</mi><mo>=</mo><mn>3</mn><mi>p</mi><mo>,</mo><mo> </mo><mtext>gradient</mtext><mo>=</mo><mfrac><mrow><mn>3</mn><mo>-</mo><mfenced><mrow><mstyle displaystyle="true"><mfrac><mrow><mn>2</mn><mi>k</mi></mrow><mi>p</mi></mfrac></mstyle><mo>+</mo><mn>3</mn></mrow></mfenced></mrow><mrow><mfenced><mrow><mn>2</mn><mi>p</mi><mo>+</mo><mn>4</mn></mrow></mfenced><mo>-</mo><mn>4</mn></mrow></mfrac><mo>,</mo><mo> </mo><mfrac><mrow><mo>-</mo><mn>6</mn></mrow><mfenced><mstyle displaystyle="true"><mfrac><mrow><mn>2</mn><mi>k</mi></mrow><mn>3</mn></mfrac></mstyle></mfenced></mfrac><mo>,</mo><mo> </mo><mo>-</mo><mfrac><mn>9</mn><mi>k</mi></mfrac></math></p>
<p>correct expression for gradient (in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>)<em><strong> (A1)</strong></em></p>
<p><em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>-</mo><mn>6</mn></mrow><mrow><mn>2</mn><mi>p</mi></mrow></mfrac><mo>,</mo><mo> </mo><mfrac><mrow><mn>9</mn><mo>-</mo><mn>3</mn></mrow><mrow><mn>4</mn><mo>-</mo><mfenced><mrow><mn>2</mn><mi>p</mi><mo>+</mo><mn>4</mn></mrow></mfenced></mrow></mfrac><mo>,</mo><mo> </mo><mo>-</mo><mfrac><mrow><mn>3</mn><mi>p</mi></mrow><msup><mi>p</mi><mn>2</mn></msup></mfrac><mo>,</mo><mo> </mo><mfrac><mrow><mn>3</mn><mo>-</mo><mfenced><mrow><mstyle displaystyle="true"><mfrac><mrow><mn>2</mn><mfenced><mrow><mn>3</mn><mi>p</mi></mrow></mfenced></mrow><mi>p</mi></mfrac></mstyle><mo>+</mo><mn>3</mn></mrow></mfenced></mrow><mrow><mfenced><mrow><mn>2</mn><mi>p</mi><mo>+</mo><mn>4</mn></mrow></mfenced><mstyle displaystyle="true"><mo>-</mo></mstyle><mstyle displaystyle="true"><mn>4</mn></mstyle></mrow></mfrac><mo>,</mo><mo> </mo><mo>-</mo><mfrac><mn>9</mn><mrow><mn>3</mn><mi>p</mi></mrow></mfrac></math></p>
<p>gradient of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>2</mn></msub></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>3</mn><mi>p</mi></mfrac><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mo>-</mo><mn>3</mn><msup><mi>p</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced></math> <em><strong> A1 N3</strong></em></p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Irina uses a set of coordinate axes to draw her design of a window. The base of the window is on the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis, the upper part of the window is in the form of a quadratic curve and the sides are vertical lines, as shown on the diagram. The curve has end points <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>10</mn><mo>)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>8</mn><mo>,</mo><mo> </mo><mn>10</mn><mo>)</mo></math> and its vertex is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>4</mn><mo>,</mo><mo> </mo><mn>12</mn><mo>)</mo></math>. Distances are measured in centimetres.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p>The quadratic curve can be expressed in the form <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> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>8</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence form two equations in terms of <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>.</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>Hence find the equation of the quadratic curve.</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>Find the area of the shaded region in Irina’s design.</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>10</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>64</mn><mi>a</mi><mo>+</mo><mn>8</mn><mi>b</mi><mo>+</mo><mn>10</mn><mo>=</mo><mn>10</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>16</mn><mi>a</mi><mo>+</mo><mn>4</mn><mi>b</mi><mo>+</mo><mn>10</mn><mo>=</mo><mn>12</mn><mo> </mo></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>A1</strong></em> for each equivalent expression or <em><strong>A1</strong></em> for the use of the axis of symmetry formula to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>=</mo><mfrac><mrow><mo>-</mo><mi>b</mi></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></math> or from use of derivative. Award <em><strong>A0A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>64</mn><mi>a</mi><mo>+</mo><mn>8</mn><mi>b</mi><mo>+</mo><mi>c</mi><mo>=</mo><mn>10</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>16</mn><mi>a</mi><mo>+</mo><mn>4</mn><mi>b</mi><mo>+</mo><mi>c</mi><mo>=</mo><mn>12</mn><mo> </mo></math>.<br><br></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><mo>-</mo><mfrac><mn>1</mn><mn>8</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi><mo>+</mo><mn>10</mn></math> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Award<em><strong> A1A0</strong></em> if one term is incorrect, <em><strong>A0A0</strong></em> if two or more terms are incorrect. Award at most <em><strong>A1A0</strong></em> if correct <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math> values are seen but answer not expressed as an equation.<br><br></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>recognizing the need to integrate their expression <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><mo>-</mo><mfrac><mn>1</mn><mn>8</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi><mo>+</mo><mn>10</mn><mo> </mo><mo>d</mo><mi>x</mi></math> <em><strong>(A1)</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>(A1)</strong></em> for correct integral, including limits. Condone absence of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>d</mo><mi>x</mi></math>.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>90</mn><mo>.</mo><mn>7</mn><mo> </mo><msup><mtext>cm</mtext><mn>2</mn></msup><mo> </mo><mo> </mo><mfenced><mrow><mfrac><mn>272</mn><mn>3</mn></mfrac><mo>,</mo><mo> </mo><mn>90</mn><mo>.</mo><mn>6666</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Generally, the responses were good for this last question on the paper. The main issue here was to not give the two equations in part (a)(ii) with simplified coefficients of <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>. Several candidates understood what was required but left their answers with <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mn>8</mn></mfenced><mn>2</mn></msup><mi>a</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mn>4</mn></mfenced><mn>2</mn></msup><mi>a</mi></math> un-simplified and lost marks. Some candidates used the coordinates <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>,</mo><mn>0</mn></mrow></mfenced></math> to substitute in the equation with an incorrect equation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>+</mo><mi>b</mi><mo>=</mo><mn>0</mn></math>. Candidates were successful at writing the equations in part (a)(iii). In part (b), most candidates realized that they had to use integration to find the area of the shaded region and, for the most part, were able to find a correct value for the area using either the correct equation or their obtained equation from the previous part. A common error was to integrate between <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> instead of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math>.</p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Generally, the responses were good for this last question on the paper. The main issue here was to not give the two equations in part (a)(ii) with simplified coefficients of <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>. Several candidates understood what was required but left their answers with <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mn>8</mn></mfenced><mn>2</mn></msup><mi>a</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mn>4</mn></mfenced><mn>2</mn></msup><mi>a</mi></math> un-simplified and lost marks. Some candidates used the coordinates <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>,</mo><mn>0</mn></mrow></mfenced></math> to substitute in the equation with an incorrect equation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>+</mo><mi>b</mi><mo>=</mo><mn>0</mn></math>. Candidates were successful at writing the equations in part (a)(iii). In part (b), most candidates realized that they had to use integration to find the area of the shaded region and, for the most part, were able to find a correct value for the area using either the correct equation or their obtained equation from the previous part. A common error was to integrate between <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> instead of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math>.</p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Generally, the responses were good for this last question on the paper. The main issue here was to not give the two equations in part (a)(ii) with simplified coefficients of <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>. Several candidates understood what was required but left their answers with <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mn>8</mn></mfenced><mn>2</mn></msup><mi>a</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mn>4</mn></mfenced><mn>2</mn></msup><mi>a</mi></math> un-simplified and lost marks. Some candidates used the coordinates <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>,</mo><mn>0</mn></mrow></mfenced></math> to substitute in the equation with an incorrect equation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>+</mo><mi>b</mi><mo>=</mo><mn>0</mn></math>. Candidates were successful at writing the equations in part (a)(iii). In part (b), most candidates realized that they had to use integration to find the area of the shaded region and, for the most part, were able to find a correct value for the area using either the correct equation or their obtained equation from the previous part. A common error was to integrate between <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> instead of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math>.</p>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Generally, the responses were good for this last question on the paper. The main issue here was to not give the two equations in part (a)(ii) with simplified coefficients of <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>. Several candidates understood what was required but left their answers with <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mn>8</mn></mfenced><mn>2</mn></msup><mi>a</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mn>4</mn></mfenced><mn>2</mn></msup><mi>a</mi></math> un-simplified and lost marks. Some candidates used the coordinates <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>,</mo><mn>0</mn></mrow></mfenced></math> to substitute in the equation with an incorrect equation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>+</mo><mi>b</mi><mo>=</mo><mn>0</mn></math>. Candidates were successful at writing the equations in part (a)(iii). In part (b), most candidates realized that they had to use integration to find the area of the shaded region and, for the most part, were able to find a correct value for the area using either the correct equation or their obtained equation from the previous part. A common error was to integrate between <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> instead of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math>.</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="text-align: left;">Inspectors are investigating the carbon dioxide emissions of a power plant. Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math> be the rate, in tonnes per hour, at which carbon dioxide is being emitted and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> be the time in hours since the inspection began.</p>
<p>When <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math> is plotted against <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>, the total amount of carbon dioxide produced is represented by the area between the graph and the horizontal <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>-axis.</p>
<p>The rate, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math>, is measured over the course of two hours. The results are shown in the following table.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p> </p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the trapezoidal rule with an interval width of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>4</mn></math> to estimate the total amount of carbon dioxide emitted during these two hours.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The real amount of carbon dioxide emitted during these two hours was <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>72</mn></math> tonnes.</p>
<p>Find the percentage error of the estimate found in part (a).</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>attempt at using trapezoidal rule formula <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mfrac><mrow><mn>2</mn><mo>-</mo><mn>0</mn></mrow><mn>5</mn></mfrac></mfenced><mfenced><mrow><mn>30</mn><mo>+</mo><mn>50</mn><mo>+</mo><mn>2</mn><mfenced><mrow><mn>50</mn><mo>+</mo><mn>60</mn><mo>+</mo><mn>40</mn><mo>+</mo><mn>20</mn></mrow></mfenced></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p>(total carbon =) <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>84</mn></math> tonnes <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"><mfenced open="|" close="|"><mfrac><mrow><mn>84</mn><mo>-</mo><mn>72</mn></mrow><mn>72</mn></mfrac></mfenced><mo>×</mo><mn>100</mn><mo>%</mo></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for correct substitution of final answer in part (a) into percentage error formula.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>16</mn><mo>.</mo><mn>7</mn><mo>%</mo><mo> </mo><mo> </mo><mfenced><mrow><mn>16</mn><mo>.</mo><mn>6666</mn><mo>…</mo><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">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Although there were successful attempts at using the trapezoidal rule formula, there was quite a bit of confusion among candidates as to which values were to be substituted. It seemed that a significant number of candidates were approaching it with some confusion due to a lack of practice of using the trapezoidal rule formula. Calculation of percentage error in part (b) was generally well done by most candidates.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Although there were successful attempts at using the trapezoidal rule formula, there was quite a bit of confusion among candidates as to which values were to be substituted. It seemed that a significant number of candidates were approaching it with some confusion due to a lack of practice of using the trapezoidal rule formula. Calculation of percentage error in part (b) was generally well done by most candidates.</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>In an international competition, participants can answer questions in <strong>only one</strong> of the three following languages: Portuguese, Mandarin or Hindi. 80 participants took part in the competition. The number of participants answering in Portuguese, Mandarin or Hindi is shown in the table.</p>
<p style="text-align: center;"><img 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"></p>
</div>
<div class="specification">
<p>A boy is chosen at random.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the number of boys who answered questions in Portuguese.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that the boy answered questions in Hindi.</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>Two girls are selected at random.</p>
<p>Calculate the probability that one girl answered questions in Mandarin and the other answered questions in Hindi.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p>20 <em><strong>(A1) (C1)</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage">null</span> <em><strong>(A1)(A1) (C2)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(A1)</strong></em> for correct numerator, <em><strong>(A1)</strong></em> for correct denominator.</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{7}{{37}} \times \frac{{12}}{{36}} + \frac{{12}}{{37}} \times \frac{7}{{36}}"> <mfrac> <mn>7</mn> <mrow> <mn>37</mn> </mrow> </mfrac> <mo>×</mo> <mfrac> <mrow> <mn>12</mn> </mrow> <mrow> <mn>36</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>12</mn> </mrow> <mrow> <mn>37</mn> </mrow> </mfrac> <mo>×</mo> <mfrac> <mn>7</mn> <mrow> <mn>36</mn> </mrow> </mfrac> </math></span> <em><strong>(A1)(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(A1)</strong></em> for first or second correct product seen, <em><strong>(M1)</strong></em> for adding their two products or for multiplying their product by two.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{14}}{{111}}\,\,\left( {\,0.12612 \ldots ,\,\,12.6126\,{\text{% }}} \right)"> <mo>=</mo> <mfrac> <mrow> <mn>14</mn> </mrow> <mrow> <mn>111</mn> </mrow> </mfrac> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mspace width="thinmathspace"></mspace> <mn>0.12612</mn> <mo>…</mo> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mn>12.6126</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>% </mtext> </mrow> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>(A1) (C3)</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Maria owns a cheese factory. The amount of cheese, in kilograms, Maria sells in one week, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="Q">
<mi>Q</mi>
</math></span>, is given by</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="Q = 882 - 45p">
<mi>Q</mi>
<mo>=</mo>
<mn>882</mn>
<mo>−<!-- − --></mo>
<mn>45</mn>
<mi>p</mi>
</math></span>,</p>
<p>where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span> is the price of a kilogram of cheese in euros (EUR).</p>
</div>
<div class="specification">
<p>Maria earns <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(p - 6.80){\text{ EUR}}">
<mo stretchy="false">(</mo>
<mi>p</mi>
<mo>−<!-- − --></mo>
<mn>6.80</mn>
<mo stretchy="false">)</mo>
<mrow>
<mtext> EUR</mtext>
</mrow>
</math></span> for each kilogram of cheese sold.</p>
</div>
<div class="specification">
<p>To calculate her weekly profit <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="W">
<mi>W</mi>
</math></span>, in EUR, Maria multiplies the amount of cheese she sells by the amount she earns per kilogram.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down how many kilograms of cheese Maria sells in one week if the price of a kilogram of cheese is 8 EUR.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find how much Maria earns in one week, from selling cheese, if the price of a kilogram of cheese is 8 EUR.</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 an expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="W">
<mi>W</mi>
</math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the price, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span>, that will give Maria the highest weekly profit.</p>
<div class="marks">[2]</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>522 (kg) <strong><em>(A1) (C1)</em></strong></p>
<p><strong><em>[1 mark]</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="522(8 - 6.80)">
<mn>522</mn>
<mo stretchy="false">(</mo>
<mn>8</mn>
<mo>−</mo>
<mn>6.80</mn>
<mo stretchy="false">)</mo>
</math></span> or equivalent <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(M1) </em></strong>for multiplying their answer to part (a) by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(8 - 6.80)">
<mo stretchy="false">(</mo>
<mn>8</mn>
<mo>−</mo>
<mn>6.80</mn>
<mo stretchy="false">)</mo>
</math></span>.</p>
<p> </p>
<p>626 (EUR) (626.40) <strong><em>(A1)</em>(ft) <em>(C2)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Follow through from part (a).</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(W = ){\text{ }}(882 - 45p)(p - 6.80)">
<mo stretchy="false">(</mo>
<mi>W</mi>
<mo>=</mo>
<mo stretchy="false">)</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mn>882</mn>
<mo>−</mo>
<mn>45</mn>
<mi>p</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>p</mi>
<mo>−</mo>
<mn>6.80</mn>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(A1)</em></strong></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(W = ) - 45{p^2} + 1188p - 5997.6">
<mo stretchy="false">(</mo>
<mi>W</mi>
<mo>=</mo>
<mo stretchy="false">)</mo>
<mo>−</mo>
<mn>45</mn>
<mrow>
<msup>
<mi>p</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1188</mn>
<mi>p</mi>
<mo>−</mo>
<mn>5997.6</mn>
</math></span> <strong><em>(A1) (C1)</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>sketch of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="W">
<mi>W</mi>
</math></span> with some indication of the maximum <strong><em>(M1)</em></strong></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 90p + 1188 = 0">
<mo>−</mo>
<mn>90</mn>
<mi>p</mi>
<mo>+</mo>
<mn>1188</mn>
<mo>=</mo>
<mn>0</mn>
</math></span> <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(M1) </em></strong>for equating the correct derivative of their part (c) to zero.</p>
<p> </p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(p = ){\text{ }}\frac{{ - 1188}}{{2 \times ( - 45)}}">
<mo stretchy="false">(</mo>
<mi>p</mi>
<mo>=</mo>
<mo stretchy="false">)</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mfrac>
<mrow>
<mo>−</mo>
<mn>1188</mn>
</mrow>
<mrow>
<mn>2</mn>
<mo>×</mo>
<mo stretchy="false">(</mo>
<mo>−</mo>
<mn>45</mn>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</math></span> <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(M1) </em></strong>for correct substitution into the formula for axis of symmetry.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(p = ){\text{ }}13.2{\text{ (EUR)}}">
<mo stretchy="false">(</mo>
<mi>p</mi>
<mo>=</mo>
<mo stretchy="false">)</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>13.2</mn>
<mrow>
<mtext> (EUR)</mtext>
</mrow>
</math></span> <strong><em>(A1)</em>(ft) <em>(C2)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Follow through from their part (c), if the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span> is such that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="6.80 < p < 19.6">
<mn>6.80</mn>
<mo><</mo>
<mi>p</mi>
<mo><</mo>
<mn>19.6</mn>
</math></span>.</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the curve <em>y</em> = 5<em>x</em><sup>3 </sup>− 3<em>x</em>.</p>
</div>
<div class="specification">
<p>The curve has a tangent at the point P(−1, −2).</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>.</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 gradient of this tangent at point P.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of this tangent. Give your answer in the form <em>y</em> = <em>mx</em> + <em>c</em>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>15<em>x</em><sup>2</sup> − 3<em><strong> (A1)(A1) (C2)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(A1)</strong></em> for 15<em>x</em><sup>2</sup>, <em><strong>(A1)</strong></em> for −3. Award at most <em><strong>(A1)</strong></em><em><strong>(A0)</strong></em> if additional terms are seen.</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>15 (−1)<sup>2</sup> − 3<em><strong> (M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong> (M1)</strong></em> for substituting −1 into their <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
</math></span>.</p>
<p> </p>
<p>= 12 <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong> (C2)</strong></em></p>
<p><strong>Note:</strong> Follow through from part (a).</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>(<em>y</em> − (−2)) = 12 (<em>x</em> − (−1)) <em><strong> (M1)</strong></em></p>
<p><strong>OR</strong></p>
<p>−2 = 12(−1) + c <em><strong> (M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong> (M1)</strong></em> for point <strong>and</strong> their gradient substituted into the equation of a line.</p>
<p> </p>
<p><em>y</em> = 12<em>x</em> + 10 <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong> (C2)</strong></em></p>
<p><strong>Note:</strong> Follow through from part (b).</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>A potter sells <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> vases per month.</p>
<p>His monthly profit in Australian dollars (AUD) can be modelled by</p>
<p><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="P\left( x \right) = - \frac{1}{5}{x^3} + 7{x^2} - 120{\text{,}}\,\,x \geqslant 0.">
<mi>P</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mfrac>
<mn>1</mn>
<mn>5</mn>
</mfrac>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>7</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>120</mn>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>⩾<!-- ⩾ --></mo>
<mn>0.</mn>
</math></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P">
<mi>P</mi>
</math></span> if no vases are sold.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Differentiate <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P\left( x \right)">
<mi>P</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span>.</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>−120 (AUD) <em><strong>(A1) (C1)</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \frac{3}{5}{x^2} + 14x">
<mo>−</mo>
<mfrac>
<mn>3</mn>
<mn>5</mn>
</mfrac>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>14</mn>
<mi>x</mi>
</math></span> <em><strong>(A1)(A1) (C2)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(A1)</strong></em> for each correct term. Award at most <em><strong>(A1)(A0)</strong></em> for extra terms seen.</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = \frac{{{x^4}}}{4}">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
</mrow>
<mn>4</mn>
</mfrac>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <em>f'</em>(<em>x</em>)</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the gradient of the graph of <em>f</em> at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = - \frac{1}{2}">
<mi>x</mi>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the <em>x</em>-coordinate of the point at which the <strong>normal</strong> to the graph of <em>f</em> has gradient <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{ - \frac{1}{8}}">
<mrow>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>8</mn>
</mfrac>
</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 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><em>x</em><sup>3</sup> <em><strong>(A1) (C1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(A0)</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{4{x^3}}}{4}">
<mfrac>
<mrow>
<mn>4</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mn>4</mn>
</mfrac>
</math></span> and not simplified to <em>x</em><sup>3</sup>.</p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( { - \frac{1}{2}} \right)^3}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>3</mn>
</msup>
</mrow>
</math></span> <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for correct substitution of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{ - \frac{1}{2}}">
<mrow>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</math></span> into their derivative.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{ - \frac{1}{8}}">
<mrow>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>8</mn>
</mfrac>
</mrow>
</math></span> (−0.125) <em><strong> (A1)</strong></em><strong>(ft)</strong><em><strong> (C2)</strong></em></p>
<p><strong>Note:</strong> Follow through from their part (a).</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><em>x</em><sup>3</sup> = 8 <em><strong>(A1)(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(A1)</strong></em> for 8 seen maybe seen as part of an equation <em>y</em> = 8<em>x</em> + <em>c</em>, <em><strong>(M1)</strong></em> for equating their derivative to 8.</p>
<p>(<em>x</em> =) 2 <em><strong>(A1) (C3)</strong></em></p>
<p><strong>Note:</strong> Do not accept (2, 4).</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The diagram shows a circular horizontal board divided into six equal sectors. The sectors are labelled white (W), yellow (Y) and blue (B).</p>
<p style="text-align: center;"><img 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"></p>
<p>A pointer is pinned to the centre of the board. The pointer is to be spun and when it stops the colour of the sector on which the pointer stops is recorded. The pointer is equally likely to stop on any of the six sectors.</p>
<p>Eva will spin the pointer twice. The following tree diagram shows all the possible outcomes.</p>
<p style="text-align: center;"><img 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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that both spins are yellow.</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 probability that at least one of the spins is yellow.</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>Write down the probability that the second spin is yellow, given that the first spin is blue.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p style="text-align: left;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{3} \times \frac{1}{3}">
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</math></span> <strong>OR </strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {\frac{1}{3}} \right)^2}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span> <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for multiplying correct probabilities.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{9}">
<mfrac>
<mn>1</mn>
<mn>9</mn>
</mfrac>
</math></span> (0.111, 0.111111…, 11.1%) <em><strong>(A1) (C2)</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 style="text-align: left;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\frac{1}{2} \times \frac{1}{3}} \right) + \left( {\frac{1}{6} \times \frac{1}{3}} \right) + \frac{1}{3}">
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>6</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</math></span> <em><strong>(M1)(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\frac{1}{2} \times \frac{1}{3}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\frac{1}{6} \times \frac{1}{3}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>6</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> or equivalent, and <em><strong>(M1)</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{3}">
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</math></span> <strong>and </strong>adding only the three correct probabilities.</p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 - {\left( {\frac{2}{3}} \right)^2}">
<mn>1</mn>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span> <em><strong>(M1)(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\frac{2}{3}}">
<mrow>
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
</mrow>
</math></span> seen and <em><strong>(M1)</strong></em> for subtracting <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {\frac{2}{3}} \right)^2}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span> from 1. This may be shown in a tree diagram with “yellow” and “not yellow” branches.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{5}{9}">
<mfrac>
<mn>5</mn>
<mn>9</mn>
</mfrac>
</math></span> (0.556, 0.555555…, 55.6%) <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong> (C3)</strong></em></p>
<p><strong>Note: </strong>Follow through marks may be awarded if their answer to part (a) is used in a correct calculation.</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 style="text-align: left;"> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{3}">
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</math></span> (0.333, 0.333333…, 33.3%) <em><strong>(A1)</strong></em><em><strong> (C1)</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msqrt><mn>12</mn><mo>-</mo><mn>2</mn><mi>x</mi></msqrt><mo>,</mo><mo> </mo><mi>x</mi><mo>≤</mo><mi>a</mi></math>. The following diagram shows part of the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math>.</p>
<p>The shaded region is enclosed by the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math>, the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis and the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p>The graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> intersects the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>a</mi><mo>,</mo><mo> </mo><mn>0</mn></mrow></mfenced></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>.</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 volume of the solid formed when the shaded region is revolved <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>360</mn><mo>°</mo></math> about the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>recognize <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>0</mn></math> <em><strong>(M1)</strong></em></p>
<p><em>eg </em> <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>12</mn><mo>-</mo><mn>2</mn><mi>x</mi></msqrt><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mo> </mo><mn>2</mn><mi>x</mi><mo>=</mo><mn>12</mn></math> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>6</mn></math> (accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>6</mn></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>6</mn><mo>,</mo><mo> </mo><mn>0</mn></mrow></mfenced></math>) <em><strong>A1 N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em> </p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to substitute either <strong>their</strong> limits or the function into volume formula (must involve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><msup><mo> </mo><mn>2</mn></msup></math>) <em><strong>(M1)</strong></em></p>
<p><em>eg </em> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mn>6</mn></msubsup><mi>f</mi><msup><mo> </mo><mn>2</mn></msup><mo>d</mo><mi>x</mi><mo> </mo><mo>,</mo><mo> </mo><mi mathvariant="normal">π</mi><mo>∫</mo><msup><mfenced><msqrt><mn>12</mn><mo>-</mo><mn>2</mn><mi>x</mi></msqrt></mfenced><mn>2</mn></msup><mo> </mo><mo>,</mo><mo> </mo><mi mathvariant="normal">π</mi><msubsup><mo>∫</mo><mn>0</mn><mn>6</mn></msubsup><mn>12</mn><mo>-</mo><mn>2</mn><mi>x</mi><mo> </mo><mo>d</mo><mi>x</mi></math> </p>
<p>correct integration of each term <em><strong>A1 A1</strong></em></p>
<p><em>eg </em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn><mi>x</mi><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><mo>,</mo><mo> </mo><mn>12</mn><mi>x</mi><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>c</mi><mo> </mo><mo>,</mo><mo> </mo><msubsup><mfenced open="[" close="]"><mrow><mn>12</mn><mi>x</mi><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mn>0</mn><mn>6</mn></msubsup></math></p>
<p>substituting limits into <strong>their integrated</strong> function and subtracting (in any order) <em><strong>(M1)</strong></em></p>
<p><em>eg </em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">π</mi><mfenced><mrow><mn>12</mn><mfenced><mn>6</mn></mfenced><mo>-</mo><msup><mfenced><mn>6</mn></mfenced><mn>2</mn></msup></mrow></mfenced><mo>-</mo><mi mathvariant="normal">π</mi><mfenced><mn>0</mn></mfenced><mo> </mo><mo>,</mo><mo> </mo><mn>72</mn><mi mathvariant="normal">π</mi><mo>-</mo><mn>36</mn><mi mathvariant="normal">π</mi><mo> </mo><mo>,</mo><mo> </mo><mfenced><mrow><mn>12</mn><mfenced><mn>6</mn></mfenced><mo>-</mo><msup><mfenced><mn>6</mn></mfenced><mn>2</mn></msup></mrow></mfenced><mo>-</mo><mfenced><mn>0</mn></mfenced></math></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>M0</strong></em> if candidate has substituted into <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>,</mo><mo> </mo><mi>f</mi><msup><mo> </mo><mn>2</mn></msup></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo></math>.</p>
<p> </p>
<p>volume<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>36</mn><mi mathvariant="normal">π</mi></math> <em><strong>A1 N2</strong></em> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A factory produces shirts. The cost, <em>C</em>, in Fijian dollars (FJD), of producing<em> x</em> shirts can be modelled by</p>
<p style="text-align: center;"><em>C</em>(<em>x</em>) = (<em>x</em> − 75)<sup>2</sup> + 100.</p>
</div>
<div class="specification">
<p>The cost of production should not exceed 500 FJD. To do this the factory needs to produce at least 55 shirts and at most <em>s</em> shirts.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the cost of producing 70 shirts.</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 value of <em>s</em>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the number of shirts produced when the cost of production is lowest.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>(70 − 75)<sup>2</sup> + 100 <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for substituting in <em>x</em> = 70.</p>
<p>125 <em><strong>(A1) (C2)</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>(<em>s</em> − 75)<sup>2</sup> + 100 = 500 <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for equating <em>C</em>(<em>x</em>) to 500. Accept an inequality instead of =.</p>
<p><strong>OR</strong></p>
<p><img src="data:image/png;base64,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"> <em><strong>(M1)</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for sketching correct graph(s).</p>
<p>(<em>s</em> =) 95 <em><strong>(A1) (C2)</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img 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"> <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for an attempt at finding the minimum point using graph.</p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{95 + 55}}{2}">
<mfrac>
<mrow>
<mn>95</mn>
<mo>+</mo>
<mn>55</mn>
</mrow>
<mn>2</mn>
</mfrac>
</math></span> <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for attempting to find the mid-point between their part (b) and 55.</p>
<p><strong>OR</strong></p>
<p>(<em>C'</em>(<em>x</em>) =) 2<em>x</em> − 150 = 0 <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for an attempt at differentiation that is correctly equated to zero.</p>
<p>75 <em><strong>(A1) (C2)</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>2</mn></math>.</p>
</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"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the normal to the curve at the point where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>y</mi><mo>-</mo><mi>x</mi><mo>+</mo><mn>3</mn><mo>=</mo><mn>0</mn></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p 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>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>4</mn></math> <strong>A1</strong></p>
<p> </p>
<p><strong>[1 mark]</strong></p>
<div class="question_part_label">a.</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>1</mn></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn></math> <strong>M1</strong></p>
<p>Gradient of normal is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac></math> <strong>A1</strong></p>
<p>When <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>1</mn><mo>-</mo><mn>4</mn><mo>+</mo><mn>2</mn><mo>=</mo><mo>-</mo><mn>1</mn></math> <strong>(M1)A1</strong></p>
<p> </p>
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>+</mo><mn>1</mn><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced></math> <strong>M1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>y</mi><mo>+</mo><mn>2</mn><mo>=</mo><mi>x</mi><mo>-</mo><mn>1</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>+</mo><mn>1</mn><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math> <strong>A1</strong></p>
<p> </p>
<p><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1</mn><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mn>1</mn><mo>+</mo><mi>c</mi></math> <strong>M1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mo>-</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></math> <strong>A1</strong></p>
<p> </p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>y</mi><mo>-</mo><mi>x</mi><mo>+</mo><mn>3</mn><mo>=</mo><mn>0</mn></math> <strong>AG</strong></p>
<p> </p>
<p><strong>[6 marks]</strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the graph of the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mfrac><mi>k</mi><mi>x</mi></mfrac></math>.</p>
</div>
<div class="specification">
<p>The equation of the tangent to the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mn>2</mn></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>y</mi><mo>=</mo><mn>4</mn><mo>-</mo><mn>5</mn><mi>x</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>′</mo><mo>(</mo><mi>x</mi><mo>)</mo></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>Write down the gradient of this tangent.</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 value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">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><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>x</mi><mo>+</mo><mfrac><mi>k</mi><msup><mi>x</mi><mn>2</mn></msup></mfrac></math> <em><strong>(A1)(A1)(A1) (C3)<br></strong></em></p>
<p><strong><br></strong><strong>Note:</strong> Award <em><strong>(A1)</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>x</mi></math>, <em><strong>(A1)</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>+</mo><mi>k</mi></math>, and <em><strong>(A1)</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><msup><mi>x</mi><mn>2</mn></msup></mfrac></math>.<br>Award at most <em><strong>(A1)(A1)(A0)</strong></em> if additional terms are seen.</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"><mo>-</mo><mn>2</mn><mo>.</mo><mn>5</mn><mo> </mo><mo> </mo><mfenced><mfrac><mrow><mo>-</mo><mn>5</mn></mrow><mn>2</mn></mfrac></mfenced></math> <em><strong>(A1) (C1)<br></strong></em></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"><mo>-</mo><mn>2</mn><mo>.</mo><mn>5</mn><mo>=</mo><mn>2</mn><mo>×</mo><mfenced><mrow><mo>-</mo><mn>2</mn></mrow></mfenced><mo>+</mo><mfrac><mi>k</mi><msup><mfenced><mrow><mo>-</mo><mn>2</mn></mrow></mfenced><mn>2</mn></msup></mfrac></math> <em><strong>(M1)<br></strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>(M1)</strong></em> for equating their gradient from part (b) to their substituted derivative from part (a).</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>k</mi><mo>=</mo></mrow></mfenced><mo> </mo><mn>6</mn></math> <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong> (C2)</strong></em></p>
<p><strong><br>Note:</strong> Follow through from parts (a) and (b).</p>
<p><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>A function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is given by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = 4{x^3} + \frac{3}{{{x^2}}} - 3,{\text{ }}x \ne 0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>4</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mfrac>
<mn>3</mn>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mo>−<!-- − --></mo>
<mn>3</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>≠<!-- ≠ --></mo>
<mn>0</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the derivative of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find 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> at which the gradient of the tangent is equal to 6.</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="12{x^2} - \frac{6}{{{x^3}}}">
<mn>12</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mfrac>
<mn>6</mn>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span> or equivalent <strong><em>(A1)(A1)(A1) (C3)</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="12{x^2}">
<mn>12</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span>, <strong><em>(A1) </em></strong>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 6">
<mo>−</mo>
<mn>6</mn>
</math></span> and <strong><em>(A1) </em></strong>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{{{x^3}}}">
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^{ - 3}}">
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</msup>
</mrow>
</math></span>. Award at most <strong><em>(A1)(A1)(A0) </em></strong>if additional terms seen.</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="12{x^2} - \frac{6}{{{x^3}}} = 6">
<mn>12</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mfrac>
<mn>6</mn>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mo>=</mo>
<mn>6</mn>
</math></span> <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(M1) </em></strong>for equating their derivative to 6.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(1,{\text{ }}4)">
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>4</mn>
<mo stretchy="false">)</mo>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><strong>OR</strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1,{\text{ }}y = 4">
<mi>x</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>y</mi>
<mo>=</mo>
<mn>4</mn>
</math></span> <strong><em>(A1)</em>(ft)<em>(A1)</em>(ft) <em>(C3)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>A frequent wrong answer seen in scripts is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(1,{\text{ }}6)">
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>6</mn>
<mo stretchy="false">)</mo>
</math></span> for this answer with correct working award <strong><em>(M1)(A0)(A1) </em></strong>and if there is no working award <strong><em>(C1)</em></strong>.</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>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = {x^3} - 2{x^2} + ax + 6">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>2</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>a</mi>
<mi>x</mi>
<mo>+</mo>
<mn>6</mn>
</math></span>. Part of the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is shown in 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 graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> crosses the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-axis at the point P. The line <em>L</em> is tangent to the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> at P.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the coordinates of P.</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 <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">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, find the equation of <em>L</em> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> has a local minimum at the point Q. The line <em>L</em> passes through Q.</p>
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>.</p>
<div class="marks">[8]</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>valid approach <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( 0 \right)">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{0^3} - 2{\left( 0 \right)^2} + a\left( 0 \right) + 6">
<mrow>
<msup>
<mn>0</mn>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>a</mi>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mn>6</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( 0 \right) = 6">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>6</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {0,\,\,y} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mn>0</mn>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p>(0, 6) (accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> = 0 <strong>and </strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span> = 6) <em><strong> A1 N2</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f' = 3{x^2} - 4x + a">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mo>=</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>4</mn>
<mi>x</mi>
<mo>+</mo>
<mi>a</mi>
</math></span> <em><strong> A2 N2</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>valid approach <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( 0 \right)">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
</math></span></p>
<p>correct working <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3{\left( 0 \right)^2} - 4\left( 0 \right) + a">
<mn>3</mn>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>4</mn>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>a</mi>
</math></span>, slope = <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="f'\left( 0 \right) = a">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>a</mi>
</math></span></p>
<p>attempt to substitute gradient and coordinates into linear equation <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y - 6 = a\left( {x - 0} \right)">
<mi>y</mi>
<mo>−</mo>
<mn>6</mn>
<mo>=</mo>
<mi>a</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>0</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y - 0 = a\left( {x - 6} \right)">
<mi>y</mi>
<mo>−</mo>
<mn>0</mn>
<mo>=</mo>
<mi>a</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>6</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="6 = a\left( 0 \right) + c">
<mn>6</mn>
<mo>=</mo>
<mi>a</mi>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>c</mi>
</math></span>, <em>L</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = ax + 6">
<mo>=</mo>
<mi>a</mi>
<mi>x</mi>
<mo>+</mo>
<mn>6</mn>
</math></span></p>
<p>correct equation <em><strong>A1 N3</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = ax + 6">
<mi>y</mi>
<mo>=</mo>
<mi>a</mi>
<mi>x</mi>
<mo>+</mo>
<mn>6</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y - 6 = ax">
<mi>y</mi>
<mo>−</mo>
<mn>6</mn>
<mo>=</mo>
<mi>a</mi>
<mi>x</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y - 6 = a\left( {x - 0} \right)">
<mi>y</mi>
<mo>−</mo>
<mn>6</mn>
<mo>=</mo>
<mi>a</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>0</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid approach to find intersection <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = L">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>L</mi>
</math></span></p>
<p>correct equation<em><strong> (A1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^3} - 2{x^2} + ax + 6 = ax + 6">
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>a</mi>
<mi>x</mi>
<mo>+</mo>
<mn>6</mn>
<mo>=</mo>
<mi>a</mi>
<mi>x</mi>
<mo>+</mo>
<mn>6</mn>
</math></span></p>
<p>correct working<em><strong> (A1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^3} - 2{x^2} = 0">
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2}(x - 2) = 0">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>−</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 2">
<mi>x</mi>
<mo>=</mo>
<mn>2</mn>
</math></span> at Q <em><strong>(A1)</strong></em></p>
<p> </p>
<p>valid approach to find minimum<em><strong> (M1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right) = 0">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p>correct equation <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3{x^2} - 4x + a = 0">
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>4</mn>
<mi>x</mi>
<mo>+</mo>
<mi>a</mi>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p>substitution of <strong>their</strong> value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> at Q into <strong>their</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right) = 0">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> equation<em><strong> (M1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3{\left( 2 \right)^2} - 4\left( 2 \right) + a = 0">
<mn>3</mn>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mn>2</mn>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>4</mn>
<mrow>
<mo>(</mo>
<mn>2</mn>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>a</mi>
<mo>=</mo>
<mn>0</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="12 - 8 + a = 0">
<mn>12</mn>
<mo>−</mo>
<mn>8</mn>
<mo>+</mo>
<mi>a</mi>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> = −4 <em><strong>A1 N0</strong></em></p>
<p> </p>
<p><em><strong>[8 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 surface area of an open box with a volume of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>32</mn><mo> </mo><msup><mtext>cm</mtext><mn>3</mn></msup></math> and a square base with sides of length <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo> </mo><mtext>cm</mtext></math> is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>128</mn><mi>x</mi></mfrac></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mn>0</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mo>′</mo><mo>(</mo><mi>x</mi><mo>)</mo></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>Solve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>0</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>Interpret your answer to (b)(i) in context.</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>S</mi><mfenced><mi>x</mi></mfenced><mo>=</mo></mrow></mfenced><mo> </mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>128</mn><msup><mi>x</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for expressing second term with a negative power. This may be implied by <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><msup><mi>x</mi><mn>2</mn></msup></mfrac></math> seen as part of their answer.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>x</mi><mo>-</mo><mfrac><mn>128</mn><msup><mi>x</mi><mn>2</mn></msup></mfrac></math> <strong>OR</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>x</mi><mo>-</mo><mn>128</mn><msup><mi>x</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></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>2</mn><mi>x</mi></math> and <em><strong>A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>128</mn><msup><mi>x</mi><mn>2</mn></msup></mfrac></math>. The first <em><strong>A1</strong></em> is for <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup></math> differentiated correctly and is independent of the <em><strong>(M1)</strong></em>.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>any correct manipulation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>x</mi><mo>-</mo><mfrac><mn>128</mn><msup><mi>x</mi><mn>2</mn></msup></mfrac><mo>=</mo><mn>0</mn></math> e.g. <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>128</mn><mo>=</mo><mn>0</mn></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>OR</strong></p>
<p>sketch of graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo></math> with root indicated <em><strong>(M1)</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p>sketch of graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> with minimum indicated <em><strong>(M1)</strong></em></p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>4</mn></math> <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Value must be positive. Follow through from their part (a) irrespective of working.</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>the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> that will minimize surface area of the box <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Accept ‘optimize’ in place of minimize.</p>
<p> </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;">
<p>In part (a), many candidates scored at least the mark for correctly differentiating <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup></math> although differentiating <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>128</mn><mi>x</mi></mfrac></math> proved to be more problematic, not realizing that the term could be written as <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>128</mn><msup><mi>x</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>. Some who did realize it, made a mistake while differentiating the negative index. In part (b)(i), the manipulation of the equation was frequently incorrect; those that used their GDC got the correct answer with no working. Many candidates could follow the instruction but where errors were made in part (a), valid solutions for part (b) proved tricky with some negative values seen. In part (b)(ii), a significant number of candidates did not appreciate what is meant by a gradient function equal to zero. Of those who had some idea, the words minimize and maximize were seen but not always in terms of the surface area. Many incorrect answers referred to the volume. Many candidates had difficulty communicating an interpretation of their answer in context. This resulted in several negative answers found for part (b)(i) being left as is, when contextually, negative answers would not make sense.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>In part (a), many candidates scored at least the mark for correctly differentiating <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup></math> although differentiating <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>128</mn><mi>x</mi></mfrac></math> proved to be more problematic, not realizing that the term could be written as <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>128</mn><msup><mi>x</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>. Some who did realize it, made a mistake while differentiating the negative index. In part (b)(i), the manipulation of the equation was frequently incorrect; those that used their GDC got the correct answer with no working. Many candidates could follow the instruction but where errors were made in part (a), valid solutions for part (b) proved tricky with some negative values seen. In part (b)(ii), a significant number of candidates did not appreciate what is meant by a gradient function equal to zero. Of those who had some idea, the words minimize and maximize were seen but not always in terms of the surface area. Many incorrect answers referred to the volume. Many candidates had difficulty communicating an interpretation of their answer in context. This resulted in several negative answers found for part (b)(i) being left as is, when contextually, negative answers would not make sense.</p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>In part (a), many candidates scored at least the mark for correctly differentiating <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup></math> although differentiating <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>128</mn><mi>x</mi></mfrac></math> proved to be more problematic, not realizing that the term could be written as <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>128</mn><msup><mi>x</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>. Some who did realize it, made a mistake while differentiating the negative index. In part (b)(i), the manipulation of the equation was frequently incorrect; those that used their GDC got the correct answer with no working. Many candidates could follow the instruction but where errors were made in part (a), valid solutions for part (b) proved tricky with some negative values seen. In part (b)(ii), a significant number of candidates did not appreciate what is meant by a gradient function equal to zero. Of those who had some idea, the words minimize and maximize were seen but not always in terms of the surface area. Many incorrect answers referred to the volume. Many candidates had difficulty communicating an interpretation of their answer in context. This resulted in several negative answers found for part (b)(i) being left as is, when contextually, negative answers would not make sense.</p>
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>A company’s profit per year was found to be changing at a rate of</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mn>3</mn><msup><mi>t</mi><mn>2</mn></msup><mo>-</mo><mn>8</mn><mi>t</mi></math></p>
<p>where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> is the company’s profit in thousands of dollars and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> is the time since the company was founded, measured in years.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine whether the profit is increasing or decreasing 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">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>One year after the company was founded, the profit was <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math> thousand dollars.</p>
<p>Find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>(</mo><mi>t</mi><mo>)</mo></math>, when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>≥</mo><mn>0</mn></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>(when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>2</mn></math>)</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mn>4</mn></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo><</mo><mn>0</mn></math> (equivalent in words) <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><msup><mfenced><mn>2</mn></mfenced><mn>2</mn></msup><mo>-</mo><mn>8</mn><mfenced><mn>2</mn></mfenced><mo>=</mo><mo>-</mo><mn>4</mn></math> <em><strong>M1</strong></em></p>
<p>therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> is decreasing <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>sketch with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>2</mn></math> indicated in 4th quadrant <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>-intercepts identified <em><strong>M1</strong></em></p>
<p>therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> is decreasing <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>P</mi><mfenced><mi>t</mi></mfenced><mo>=</mo></mrow></mfenced><mo> </mo><mo> </mo><msup><mi>t</mi><mn>3</mn></msup><mo>-</mo><mn>4</mn><msup><mi>t</mi><mn>2</mn></msup><mfenced><mrow><mo>+</mo><mi>c</mi></mrow></mfenced></math> <em><strong>A1A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>=</mo><msup><mn>1</mn><mn>3</mn></msup><mo>-</mo><mn>4</mn><msup><mfenced><mn>1</mn></mfenced><mn>2</mn></msup><mo>+</mo><mi>c</mi></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>M1</strong> </em>for substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>4</mn><mo>)</mo></math> into their equation with <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>+</mo><mi>c</mi></math> seen.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>7</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><msup><mi>t</mi><mn>3</mn></msup><mo>-</mo><mn>4</mn><msup><mi>t</mi><mn>2</mn></msup><mo>+</mo><mn>7</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Even some weaker candidates were able to score in this part of the question as many candidates understood the process required to determine whether the profit is increasing or decreasing.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Many candidates failed to recognize that they needed to integrate the original function. Of the few that attempted to find the value of the constant the vast majority substituted 4000 rather than 4. So, a correct final expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>(</mo><mi>t</mi><mo>)</mo></math> was rarely seen.</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A modern art painting is contained in a square frame. The painting has a shaded region bounded by a smooth curve and a horizontal line.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p>When the painting is placed on a coordinate axes such that the bottom left corner of the painting has coordinates <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mo>−</mo><mn>1</mn><mo>,</mo><mo> </mo><mo>−</mo><mn>1</mn><mo>)</mo></math> and the top right corner 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></math>, the curve can be modelled 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> and the horizontal line can be modelled by the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis. Distances are measured in metres.</p>
</div>
<div class="specification">
<p>The artist used the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mrow><mo>-</mo><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>12</mn></mrow><mn>10</mn></mfrac></math> to draw the curve.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the trapezoidal rule, with the values given in the following table, to approximate the area of the shaded region.</p>
<p style="text-align:center;"><img src="data:image/png;base64,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"></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the exact area of the shaded region in the painting.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area of the unshaded region in the painting.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mrow><mn>0</mn><mo>.</mo><mn>6</mn><mo>+</mo><mn>0</mn><mo>+</mo><mn>2</mn><mfenced><mrow><mn>1</mn><mo>.</mo><mn>2</mn><mo>+</mo><mn>1</mn><mo>.</mo><mn>2</mn></mrow></mfenced></mrow></mfenced></math> <em><strong>(A1)(M1)</strong></em></p>
<p> <br><strong>Note:</strong> Award <em><strong>A1</strong> </em>for evidence of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mn>1</mn></math>, <em><strong>M1</strong> </em>for a correct substitution into trapezoidal rule (allow for an incorrect <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math> only). The zero can be omitted in the working.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>7</mn><mo> </mo><msup><mtext>m</mtext><mn>2</mn></msup></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mrow><mo>-</mo><mn>1</mn></mrow><mn>2</mn></msubsup><mfrac><mrow><mo>-</mo><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>12</mn></mrow><mn>10</mn></mfrac><mo>d</mo><mi>x</mi></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mrow><mo>-</mo><mn>1</mn></mrow><mn>2</mn></msubsup><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>d</mo><mi>x</mi></math> <em><strong>(M1)</strong></em></p>
<p> <br><strong>Note:</strong> Award <em><strong>M1</strong> </em>for using definite integration with correct limits.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>925</mn><mo> </mo><msup><mtext>m</mtext><mn>2</mn></msup></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note: </strong>Question requires exact answer, do not award final <em><strong>A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>93</mn></math>.</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"><mn>9</mn><mo>-</mo><mn>2</mn><mo>.</mo><mn>925</mn></math> <em><strong>(M1)</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>M1</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn></math> seen as part of a subtraction.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>6</mn><mo>.</mo><mn>08</mn><mo> </mo><msup><mtext>m</mtext><mn>2</mn></msup><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mn>6</mn><mo>.</mo><mn>075</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>There seemed a better attempt at using the trapezium rule in this session compared to the two 2021 sessions. Despite many incorrect values for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math>, candidates obtained the method mark for a correctly substituted formula (excluding <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math>).</p>
<p> </p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>The exact answer of 2.925 was asked for in the question, yet candidates frequently rounded to three significant figures and hence lost the final mark.</p>
<p> </p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Many candidates were able to correctly find the area of the unshaded region.</p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = 1 + {{\text{e}}^{ - x}}">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−<!-- − --></mo>
<mi>x</mi>
</mrow>
</msup>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g(x) = 2x + b">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mi>b</mi>
</math></span>, for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \in \mathbb{R}">
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span> is a constant.</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="(g \circ f)(x)">
<mo stretchy="false">(</mo>
<mi>g</mi>
<mo>∘</mo>
<mi>f</mi>
<mo stretchy="false">)</mo>
<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.</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="\mathop {\lim }\limits_{x \to + \infty } (g \circ f)(x) = - 3">
<munder>
<mrow>
<mo form="prefix">lim</mo>
</mrow>
<mrow>
<mi>x</mi>
<mo stretchy="false">→</mo>
<mo>+</mo>
<mi mathvariant="normal">∞</mi>
</mrow>
</munder>
<mo></mo>
<mo stretchy="false">(</mo>
<mi>g</mi>
<mo>∘</mo>
<mi>f</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>−</mo>
<mn>3</mn>
</math></span>, find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">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>attempt to form composite <strong><em>(M1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g(1 + {{\text{e}}^{ - x}})">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</math></span></p>
<p>correct function <strong><em>A1 N2</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(g \circ f)(x) = 2 + b + 2{{\text{e}}^{ - x}},{\text{ }}2(1 + {{\text{e}}^{ - x}}) + b">
<mo stretchy="false">(</mo>
<mi>g</mi>
<mo>∘</mo>
<mi>f</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>2</mn>
<mo>+</mo>
<mi>b</mi>
<mo>+</mo>
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>2</mn>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mi>b</mi>
</math></span></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>evidence of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {\lim }\limits_{x \to \infty } (2 + b + 2{{\text{e}}^{ - x}}) = 2 + b + \mathop {\lim }\limits_{x \to \infty } (2{{\text{e}}^{ - x}})">
<munder>
<mrow>
<mo form="prefix">lim</mo>
</mrow>
<mrow>
<mi>x</mi>
<mo stretchy="false">→</mo>
<mi mathvariant="normal">∞</mi>
</mrow>
</munder>
<mo></mo>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mo>+</mo>
<mi>b</mi>
<mo>+</mo>
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>2</mn>
<mo>+</mo>
<mi>b</mi>
<mo>+</mo>
<munder>
<mrow>
<mo form="prefix">lim</mo>
</mrow>
<mrow>
<mi>x</mi>
<mo stretchy="false">→</mo>
<mi mathvariant="normal">∞</mi>
</mrow>
</munder>
<mo></mo>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(M1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2 + b + 2{{\text{e}}^{ - \infty }}">
<mn>2</mn>
<mo>+</mo>
<mi>b</mi>
<mo>+</mo>
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mi mathvariant="normal">∞</mi>
</mrow>
</msup>
</mrow>
</math></span>, graph with horizontal asymptote when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \to \infty ">
<mi>x</mi>
<mo stretchy="false">→</mo>
<mi mathvariant="normal">∞</mi>
</math></span></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>M0 </em></strong>if candidate clearly has incorrect limit, such as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \to 0,{\text{ }}{{\text{e}}^\infty },{\text{ }}2{{\text{e}}^0}">
<mi>x</mi>
<mo stretchy="false">→</mo>
<mn>0</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mi mathvariant="normal">∞</mi>
</msup>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mn>0</mn>
</msup>
</mrow>
</math></span>.</p>
<p> </p>
<p>evidence that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{e}}^{ - x}} \to 0">
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mo stretchy="false">→</mo>
<mn>0</mn>
</math></span> (seen anywhere) <strong><em>(A1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {\lim }\limits_{x \to \infty } ({{\text{e}}^{ - x}}) = 0,{\text{ }}1 + {{\text{e}}^{ - x}} \to 1,{\text{ }}2(1) + b = - 3,{\text{ }}{{\text{e}}^{{\text{large negative number}}}} \to 0">
<munder>
<mrow>
<mo form="prefix">lim</mo>
</mrow>
<mrow>
<mi>x</mi>
<mo stretchy="false">→</mo>
<mi mathvariant="normal">∞</mi>
</mrow>
</munder>
<mo></mo>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mo stretchy="false">→</mo>
<mn>1</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>2</mn>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mi>b</mi>
<mo>=</mo>
<mo>−</mo>
<mn>3</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mrow>
<mtext>large negative number</mtext>
</mrow>
</mrow>
</msup>
</mrow>
<mo stretchy="false">→</mo>
<mn>0</mn>
</math></span>, graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {{\text{e}}^{ - x}}">
<mi>y</mi>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mi>x</mi>
</mrow>
</msup>
</mrow>
</math></span> or</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 2{{\text{e}}^{ - x}}">
<mi>y</mi>
<mo>=</mo>
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mi>x</mi>
</mrow>
</msup>
</mrow>
</math></span> with asymptote <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>, graph of composite function with asymptote <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = - 3">
<mi>y</mi>
<mo>=</mo>
<mo>−</mo>
<mn>3</mn>
</math></span></p>
<p>correct working <strong><em>(A1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2 + b = - 3">
<mn>2</mn>
<mo>+</mo>
<mi>b</mi>
<mo>=</mo>
<mo>−</mo>
<mn>3</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b = - 5">
<mi>b</mi>
<mo>=</mo>
<mo>−</mo>
<mn>5</mn>
</math></span> <strong><em>A1 N2</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>The graphs of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>6</mn><mo>−</mo><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mo>.</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>3</mn></math> intersect at <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>2</mn><mo>,</mo><mo> </mo><mn>4</mn><mo>)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mo>−</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>7</mn><mo>)</mo></math>, as shown in the following diagrams.</p>
<p>In <strong>diagram 1</strong>, the region enclosed by the lines <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>6</mn><mo>−</mo><mi>x</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>−</mo><mn>1</mn></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>2</mn></math> and the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis has been shaded.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" 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"></p>
</div>
<div class="specification">
<p>In <strong>diagram 2</strong>, the region enclosed by the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mo>.</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>3</mn></math>, and the lines <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>−</mo><mn>1</mn></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>2</mn></math> and the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis has been shaded.</p>
<p style="text-align: center;"><img 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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the area of the shaded region in <strong>diagram 1</strong>.</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 an integral for the area of the shaded region in<strong> diagram 2</strong>.</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>Calculate the area of this region.</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>Hence, determine the area enclosed between <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>6</mn><mo>-</mo><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mo>.</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>3</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>attempt to substitute <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>,</mo><mo> </mo><mn>4</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn></math> into area of a trapezoid formula <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>A</mi><mo>=</mo></mrow></mfenced><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mrow><mn>7</mn><mo>+</mo><mn>4</mn></mrow></mfenced><mfenced><mn>3</mn></mfenced></math></p>
<p><br><strong>OR</strong></p>
<p>given line expressed as an integral <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>A</mi><mo>=</mo></mrow></mfenced><msubsup><mo>∫</mo><mrow><mo>-</mo><mn>1</mn></mrow><mn>2</mn></msubsup><mfenced><mrow><mn>6</mn><mo>-</mo><mi>x</mi></mrow></mfenced><mo> </mo><mo>d</mo><mi>x</mi></math></p>
<p><br><strong>OR</strong></p>
<p>attempt to sum area of rectangle and area of triangle <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>A</mi><mo>=</mo></mrow></mfenced><mo> </mo><mn>4</mn><mo>×</mo><mn>3</mn><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mn>3</mn></mfenced><mfenced><mn>3</mn></mfenced></math></p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>16</mn><mo>.</mo><mn>5</mn></math> (square units) <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"><mfenced><mrow><mi>A</mi><mo>=</mo></mrow></mfenced><mo> </mo><msubsup><mo>∫</mo><mrow><mo>-</mo><mn>1</mn></mrow><mn>2</mn></msubsup><mn>1</mn><mo>.</mo><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mo>.</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>3</mn><mo> </mo><mo>d</mo><mi>x</mi></math> <em><strong>A1A1</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>A1</strong> </em>for the limits <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mn>1</mn></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>2</mn></math> in correct location. Award <em><strong>A1</strong> </em>for an integral of the quadratic function, <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>d</mo><mi>x</mi></math> must be included. Do not accept “<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>” in place of the function, given that two equations are in the question.</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn><mo>.</mo><mn>75</mn></math> (square units) <em><strong>A1</strong></em></p>
<p> </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><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>16</mn><mo>.</mo><mn>5</mn><mo>-</mo><mn>9</mn><mo>.</mo><mn>75</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>.</mo><mn>75</mn></math> (square units) <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>There were a variety of methods used or attempted – area of trapezoid, integration, area of triangle plus area of rectangle, area of large rectangle minus area of top triangle, trapezoidal rule. All these methods, except for trapezoidal rule, proved successful for candidates, with the most common being the use of integration.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>This was reasonably well done except for a few notation issues such as not including dx with their integrand. Those who attempted integration manually were not successful.</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;">
<p>Recognition that areas had to be subtracted was very evident.</p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The coordinates of point A are <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(6,{\text{ }} - 7)">
<mo stretchy="false">(</mo>
<mn>6</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo>−<!-- − --></mo>
<mn>7</mn>
<mo stretchy="false">)</mo>
</math></span> and the coordinates of point B are <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="( - 6,{\text{ }}2)">
<mo stretchy="false">(</mo>
<mo>−<!-- − --></mo>
<mn>6</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>2</mn>
<mo stretchy="false">)</mo>
</math></span>. Point M is the midpoint of AB.</p>
</div>
<div class="specification">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_1}">
<mrow>
<msub>
<mi>L</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> is the line through A and B.</p>
</div>
<div class="specification">
<p>The line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_2}">
<mrow>
<msub>
<mi>L</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span> is perpendicular to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_1}">
<mrow>
<msub>
<mi>L</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> and passes through M.</p>
</div>
<div class="question">
<p>Write down, in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = mx + c">
<mi>y</mi>
<mo>=</mo>
<mi>m</mi>
<mi>x</mi>
<mo>+</mo>
<mi>c</mi>
</math></span>, the equation of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_2}">
<mrow>
<msub>
<mi>L</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \frac{4}{3}x - \frac{5}{2}{\text{ }}(y = 1.33 \ldots x - 2.5)">
<mi>y</mi>
<mo>=</mo>
<mfrac>
<mn>4</mn>
<mn>3</mn>
</mfrac>
<mi>x</mi>
<mo>−</mo>
<mfrac>
<mn>5</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo>=</mo>
<mn>1.33</mn>
<mo>…</mo>
<mi>x</mi>
<mo>−</mo>
<mn>2.5</mn>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(A1)</em>(ft) <em>(C1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Follow through from parts (c)(i) and (a). Award <strong><em>(A0) </em></strong>if final answer is not written in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = mx + c">
<mi>y</mi>
<mo>=</mo>
<mi>m</mi>
<mi>x</mi>
<mo>+</mo>
<mi>c</mi>
</math></span>.</p>
<p><strong><em>[1 mark]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>A quadratic function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is given by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = a{x^2} + bx + c">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>a</mi>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>b</mi>
<mi>x</mi>
<mo>+</mo>
<mi>c</mi>
</math></span>. The points <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(0,{\text{ }}5)">
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>5</mn>
<mo stretchy="false">)</mo>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="( - 4,{\text{ }}5)">
<mo stretchy="false">(</mo>
<mo>−<!-- − --></mo>
<mn>4</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>5</mn>
<mo stretchy="false">)</mo>
</math></span> lie 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>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span>.</p>
</div>
<div class="specification">
<p>The <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-coordinate of the minimum of the graph is 3.</p>
</div>
<div class="question">
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> and of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \frac{b}{{2a}} = - 2">
<mo>−</mo>
<mfrac>
<mi>b</mi>
<mrow>
<mn>2</mn>
<mi>a</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mo>−</mo>
<mn>2</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a{( - 2)^2} - 2b + 5 = 3">
<mi>a</mi>
<mrow>
<mo stretchy="false">(</mo>
<mo>−</mo>
<mn>2</mn>
<msup>
<mo stretchy="false">)</mo>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mi>b</mi>
<mo>+</mo>
<mn>5</mn>
<mo>=</mo>
<mn>3</mn>
</math></span> or equivalent</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a{( - 4)^2} - 4b + 5 = 5">
<mi>a</mi>
<mrow>
<mo stretchy="false">(</mo>
<mo>−</mo>
<mn>4</mn>
<msup>
<mo stretchy="false">)</mo>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>4</mn>
<mi>b</mi>
<mo>+</mo>
<mn>5</mn>
<mo>=</mo>
<mn>5</mn>
</math></span> or equivalent</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2a( - 2) + b = 0">
<mn>2</mn>
<mi>a</mi>
<mo stretchy="false">(</mo>
<mo>−</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mi>b</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> or equivalent <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(M1) </em></strong>for two of the above equations.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = 0.5">
<mi>a</mi>
<mo>=</mo>
<mn>0.5</mn>
</math></span> <strong><em>(A1)</em>(ft)</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b = 2">
<mi>b</mi>
<mo>=</mo>
<mn>2</mn>
</math></span> <strong><em>(A1)</em>(ft) <em>(C3)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award at most <strong><em>(M1)(A1)</em>(ft)<em>(A0) </em></strong>if the answers are reversed.</p>
<p>Follow through from parts (a) and (b).</p>
<p> </p>
<p><strong><em>[3 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>Ellis designs a gift box. The top of the gift box is in the shape of a right-angled triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>GIK</mtext></math>.</p>
<p>A rectangular section <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>HIJL</mtext></math> is inscribed inside this triangle. The lengths of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>GH, JK, HL</mtext></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>LJ</mtext></math> are <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo> </mo><mtext>cm</mtext><mo>,</mo><mo> </mo><mi>q</mi><mo> </mo><mtext>cm</mtext><mo>,</mo><mo> </mo><mn>8</mn><mo> </mo><mtext>cm</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo> </mo><mtext>cm</mtext></math> respectively.</p>
<p style="text-align: center;"><img 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"></p>
<p style="text-align: left;">The area of the top of the gift box is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo> </mo><msup><mtext>cm</mtext><mn>2</mn></msup></math>.</p>
</div>
<div class="specification">
<p>Ellis wishes to find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math> that will minimize the area of the top of the gift box.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mfrac><mn>192</mn><mi>q</mi></mfrac><mo>+</mo><mn>3</mn><mi>q</mi><mo>+</mo><mn>48</mn></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>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>A</mi></mrow><mrow><mo>d</mo><mi>q</mi></mrow></mfrac></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>Write down an equation Ellis could solve to find this value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></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>Hence, or otherwise, find this value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mn>6</mn><mo>×</mo><mi>q</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mn>8</mn><mo>×</mo><mi>p</mi><mo>+</mo><mn>48</mn></math> <em><strong>OR </strong></em><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mrow><mi>p</mi><mo>+</mo><mn>6</mn></mrow></mfenced><mfenced><mrow><mi>q</mi><mo>+</mo><mn>8</mn></mrow></mfenced></math> <em><strong>OR </strong></em><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mn>3</mn><mi>q</mi><mo>+</mo><mn>4</mn><mi>p</mi><mo>+</mo><mn>48</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid attempt to link <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math>, using tangents, similar triangles or other method <em><strong>(M1)</strong></em></p>
<p>eg. <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mo> </mo><mi>θ</mi><mo>=</mo><mfrac><mn>8</mn><mi>p</mi></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mo> </mo><mi>θ</mi><mo>=</mo><mfrac><mi>q</mi><mn>6</mn></mfrac></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mo> </mo><mi>θ</mi><mo>=</mo><mfrac><mi>p</mi><mn>8</mn></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mo> </mo><mi>θ</mi><mo>=</mo><mfrac><mn>6</mn><mi>q</mi></mfrac></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>8</mn><mi>p</mi></mfrac><mo>=</mo><mfrac><mi>q</mi><mn>6</mn></mfrac></math></p>
<p>correct equation linking <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math> <em><strong>A1</strong></em></p>
<p>eg. <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mi>q</mi><mo>=</mo><mn>48</mn></math> <strong>OR</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mfrac><mn>48</mn><mi>q</mi></mfrac></math> <strong>OR</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>=</mo><mfrac><mn>48</mn><mi>p</mi></mfrac></math></p>
<p>substitute <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mfrac><mn>48</mn><mi>q</mi></mfrac></math> into a correct area expression <em><strong>M1</strong></em></p>
<p>eg. <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>A</mi><mo>=</mo></mrow></mfenced><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mn>6</mn><mo>×</mo><mi>q</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mn>8</mn><mo>×</mo><mfrac><mn>48</mn><mi>q</mi></mfrac><mo>+</mo><mn>48</mn></math> <strong>OR</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>A</mi><mo>=</mo></mrow></mfenced><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mrow><mfrac><mn>48</mn><mi>q</mi></mfrac><mo>+</mo><mn>6</mn></mrow></mfenced><mfenced><mrow><mi>q</mi><mo>+</mo><mn>8</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mn>3</mn><mi>q</mi><mo>+</mo><mfrac><mn>192</mn><mi>q</mi></mfrac><mo>+</mo><mn>48</mn></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>Note:</strong> The <em><strong>AG</strong></em> line must be seen with no incorrect, intermediate working, for the final <em><strong>M1</strong></em> to be awarded.</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>-</mo><mn>192</mn></mrow><msup><mi>q</mi><mn>2</mn></msup></mfrac><mo>+</mo><mn>3</mn></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"><mfrac><mrow><mo>-</mo><mn>192</mn></mrow><msup><mi>q</mi><mn>2</mn></msup></mfrac></math>, <em><strong>A1</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math>. Award <em><strong>A1A0</strong></em> if extra terms are seen.</p>
<p><em><strong><br>[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"><mfrac><mrow><mo>-</mo><mn>192</mn></mrow><msup><mi>q</mi><mn>2</mn></msup></mfrac><mo>+</mo><mn>3</mn><mo>=</mo><mn>0</mn></math> <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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>=</mo><mn>8</mn><mo> </mo><mtext>cm</mtext></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[1 mark]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.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.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>A cylinder with radius <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span> and height <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h">
<mi>h</mi>
</math></span> is shown in 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 sum of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h">
<mi>h</mi>
</math></span> for this cylinder is 12 cm.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down an equation for the area, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A">
<mi>A</mi>
</math></span>, of the <strong>curved</strong> surface in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</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>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}A}}{{{\text{d}}r}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>A</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>r</mi>
</mrow>
</mfrac>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span> when the area of the curved surface is maximized.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p style="text-align: left;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = 2\pi r\left( {12 - r} \right)">
<mi>A</mi>
<mo>=</mo>
<mn>2</mn>
<mi>π</mi>
<mi>r</mi>
<mrow>
<mo>(</mo>
<mrow>
<mn>12</mn>
<mo>−</mo>
<mi>r</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong>OR</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = 24\pi r - 2\pi {r^2}">
<mi>A</mi>
<mo>=</mo>
<mn>24</mn>
<mi>π</mi>
<mi>r</mi>
<mo>−</mo>
<mn>2</mn>
<mi>π</mi>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> <em><strong>(A1)(M1) (C2)</strong></em></p>
<p style="text-align: left;"><strong>Note:</strong> Award <em><strong>(A1)</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r + h = 12">
<mi>r</mi>
<mo>+</mo>
<mi>h</mi>
<mo>=</mo>
<mn>12</mn>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h = 12 - r">
<mi>h</mi>
<mo>=</mo>
<mn>12</mn>
<mo>−</mo>
<mi>r</mi>
</math></span> seen. Award <em><strong>(M1)</strong></em> for correctly substituting into curved surface area of a cylinder. Accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = 2\pi r\left( {12 - r} \right)">
<mi>A</mi>
<mo>=</mo>
<mn>2</mn>
<mi>π</mi>
<mi>r</mi>
<mrow>
<mo>(</mo>
<mrow>
<mn>12</mn>
<mo>−</mo>
<mi>r</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong>OR </strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = 24\pi r - 2\pi {r^2}">
<mi>A</mi>
<mo>=</mo>
<mn>24</mn>
<mi>π</mi>
<mi>r</mi>
<mo>−</mo>
<mn>2</mn>
<mi>π</mi>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span>.</p>
<p style="text-align: left;"><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align: left;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="24\pi - 4\pi r">
<mn>24</mn>
<mi>π</mi>
<mo>−</mo>
<mn>4</mn>
<mi>π</mi>
<mi>r</mi>
</math></span> <em><strong>(A1)</strong></em><strong>(ft)<em>(A1)</em>(ft)</strong><em><strong> (C2)</strong></em></p>
<p style="text-align: left;"><strong>Note:</strong> Award <em><strong>(A1)</strong></em><strong>(ft)</strong> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="24\pi">
<mn>24</mn>
<mi>π</mi>
</math></span> and <em><strong>(A1)</strong></em><strong>(ft)</strong> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 4\pi r">
<mo>−</mo>
<mn>4</mn>
<mi>π</mi>
<mi>r</mi>
</math></span> . Follow through from part (a). Award at most <em><strong>(A1)</strong></em><strong>(ft)<em>(A0)</em></strong> if additional terms are seen.</p>
<p style="text-align: left;"><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align: left;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="24\pi - 4\pi r = 0">
<mn>24</mn>
<mi>π</mi>
<mo>−</mo>
<mn>4</mn>
<mi>π</mi>
<mi>r</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> <strong><em>(M1)</em></strong></p>
<p style="text-align: left;"><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for setting <em>their</em> part (b) equal to zero.</p>
<p style="text-align: left;">6 (cm) <strong><em>(A1)</em>(ft)</strong><em><strong> (C2)</strong></em></p>
<p style="text-align: left;"><strong>Note:</strong> Follow through from part (b).</p>
<p style="text-align: left;"><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>A company produces and sells electric cars. The company’s profit, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math>, in thousands of dollars, changes based on the number of cars, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>, they produce per month.</p>
<p>The rate of change of their profit from producing <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> electric cars is modelled by</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mo>−</mo><mn>1</mn><mo>.</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>48</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>≥</mo><mn>0</mn></math>.</p>
<p>The company makes a profit of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>260</mn></math> (thousand dollars) when they produce <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>15</mn></math> electric cars.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The company regularly increases the number of cars it produces.</p>
<p>Describe how their profit changes if they increase production to over <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>30</mn></math> cars per month and up to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>50</mn></math> cars per month. Justify your answer.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>recognition of need to integrate (<em>eg</em> reverse power rule or integral symbol) <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>8</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>48</mn><mi>x</mi><mo> </mo><mfenced><mrow><mo>+</mo><mi>c</mi></mrow></mfenced></math> <em><strong>A1A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>260</mn><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>8</mn><mo>×</mo><msup><mfenced><mn>15</mn></mfenced><mn>2</mn></msup><mo>+</mo><mn>48</mn><mo>×</mo><mfenced><mn>15</mn></mfenced><mo>+</mo><mi>c</mi></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>M1</strong></em> for correct substitution of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>15</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mn>260</mn></math>. A constant of integration must be seen (can be implied by a correct answer).</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mo>-</mo><mn>280</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>8</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>48</mn><mi>x</mi><mo> </mo><mo>-</mo><mn>280</mn></math> <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>profit will decrease (with each new car produced) <em><strong>A1</strong></em></p>
<p><strong>EITHER</strong></p>
<p>because the profit function is decreasing / the gradient is negative / the rate of change of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> is negative <em><strong> R1</strong></em></p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>30</mn><mn>50</mn></msubsup><mo>-</mo><mn>1</mn><mo>.</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>48</mn><mo> </mo><mfenced><mrow><mo>d</mo><mi>x</mi></mrow></mfenced><mo>=</mo><mo>-</mo><mn>320</mn></math> <em><strong> R1</strong></em></p>
<p><br><strong>OR</strong></p>
<p>evidence of finding <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mfenced><mn>30</mn></mfenced><mo>=</mo><mn>440</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mfenced><mn>50</mn></mfenced><mo>=</mo><mn>120</mn></math> <em><strong> R1</strong></em></p>
<p><strong><br>Note:</strong> Award at most <em><strong>R1A0</strong></em> if <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mfenced><mn>30</mn></mfenced></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mfenced><mn>50</mn></mfenced></math> or both have incorrect values.</p>
<p><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The diagram shows part of the graph of a function <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 graph passes through point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}(1,{\text{ }}3)">
<mrow>
<mtext>A</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>3</mn>
<mo stretchy="false">)</mo>
</math></span>.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-16_om_06.22.37.png" alt="M17/5/MATSD/SP1/ENG/TZ2/13"></p>
</div>
<div class="specification">
<p>The tangent to 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> at A has equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = - 2x + 5">
<mi>y</mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>5</mn>
</math></span>. Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="N">
<mi>N</mi>
</math></span> be the normal to 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> at A.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(1)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="N">
<mi>N</mi>
</math></span>. Give your answer in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="ax + by + d = 0">
<mi>a</mi>
<mi>x</mi>
<mo>+</mo>
<mi>b</mi>
<mi>y</mi>
<mo>+</mo>
<mi>d</mi>
<mo>=</mo>
<mn>0</mn>
</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="d \in \mathbb{Z}">
<mi>d</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Draw the line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="N">
<mi>N</mi>
</math></span> on the diagram above.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>3 <strong><em>(A1)</em></strong> <strong><em>(C1)</em></strong></p>
<p> </p>
<p><strong>Notes:</strong> Accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 3">
<mi>y</mi>
<mo>=</mo>
<mn>3</mn>
</math></span></p>
<p> </p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3 = 0.5(1) + c">
<mn>3</mn>
<mo>=</mo>
<mn>0.5</mn>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mi>c</mi>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><strong>OR</strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y - 3 = 0.5(x - 1)">
<mi>y</mi>
<mo>−</mo>
<mn>3</mn>
<mo>=</mo>
<mn>0.5</mn>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(A1)(A1)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>(A1) </em></strong>for correct gradient, <strong><em>(A1) </em></strong>for correct substitution of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}(1,{\text{ }}3)">
<mrow>
<mtext>A</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>3</mn>
<mo stretchy="false">)</mo>
</math></span> in the equation of line.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x - 2y + 5 = 0">
<mi>x</mi>
<mo>−</mo>
<mn>2</mn>
<mi>y</mi>
<mo>+</mo>
<mn>5</mn>
<mo>=</mo>
<mn>0</mn>
</math></span> or any integer multiple <strong><em>(A1)</em>(ft)</strong> <strong><em>(C3)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>(A1)</em>(ft) </strong>for their equation correctly rearranged in the indicated form.</p>
<p>The candidate’s answer <strong>must </strong>be an equation for this mark.</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><img src="images/Schermafbeelding_2017-08-16_om_08.26.38.png" alt="M17/5/MATSD/SP1/ENG/TZ2/13.c/M"> <strong><em>(M1)(A1)</em>(ft)</strong> <strong><em>(C2)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>M1) </em></strong>for a straight line, with positive gradient, passing through <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(1,{\text{ }}3)">
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>3</mn>
<mo stretchy="false">)</mo>
</math></span>, <strong><em>(A1)</em>(ft) </strong>for line (or extension of their line) passing approximately through 2.5 or their intercept with the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-axis.</p>
<p> </p>
<p><strong><em>[2 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 following diagram shows part of the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = \left( {6 - 3x} \right)\left( {4 + x} \right)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>6</mn>
<mo>−<!-- − --></mo>
<mn>3</mn>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>4</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \in \mathbb{R}">
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>. The shaded region <em>R</em> is bounded by the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-axis and the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span>.</p>
<p style="text-align: center;"><img 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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down an integral for the area of region <em>R</em>.</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 area of region <em>R</em>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The three points A(0, 0) , B(3, 10) and C(<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>, 0) define the vertices of a triangle.</p>
<p><img style="display: block;margin-left:auto;margin-right:auto;" 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"></p>
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>, the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-coordinate of C, such that the area of the triangle is equal to the area of region <em>R</em>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><em>A</em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^2 {\left( {6 - 3x} \right)\left( {4 + x} \right){\text{d}}x} "> <msubsup> <mo>∫</mo> <mn>0</mn> <mn>2</mn> </msubsup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>6</mn> <mo>−</mo> <mn>3</mn> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>4</mn> <mo>+</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </math></span> <em><strong>A1A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong> </em>for the limits <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> = 0, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> = 2. Award <em><strong>A1</strong></em> for an integral of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x)"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span>.</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>28 <em><strong>A1</strong></em></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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="28 = 0.5 \times a \times 10">
<mn>28</mn>
<mo>=</mo>
<mn>0.5</mn>
<mo>×</mo>
<mi>a</mi>
<mo>×</mo>
<mn>10</mn>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="5.6\left( {\frac{{28}}{5}} \right)">
<mn>5.6</mn>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>28</mn>
</mrow>
<mn>5</mn>
</mfrac>
</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">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>It was pleasing to see that, for those candidates who made a reasonable attempt at the paper, many were able to identify the correct values on the tree diagram.</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>
<br><hr><br><div class="specification">
<p>The following diagram shows the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f’">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
</math></span>, the derivative of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span>.</p>
<p><img src="images/Schermafbeelding_2017-08-11_om_08.50.59.png" alt="M17/5/MATME/SP1/ENG/TZ1/06"></p>
<p>The graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f’">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
</math></span> has a local minimum at A, a local maximum at B and passes through <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(4,{\text{ }} - 2)">
<mo stretchy="false">(</mo>
<mn>4</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo>−<!-- − --></mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</math></span>.</p>
</div>
<div class="specification">
<p>The point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(4,{\text{ }}3)">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mn>4</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>3</mn>
<mo stretchy="false">)</mo>
</math></span> lies on the graph of the function, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the gradient of the curve of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> at P.</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 equation of the normal to the curve of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> at P.</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>Determine the concavity of the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4 < x < 5">
<mn>4</mn>
<mo><</mo>
<mi>x</mi>
<mo><</mo>
<mn>5</mn>
</math></span> <strong>and </strong>justify your answer.</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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 2">
<mo>−</mo>
<mn>2</mn>
</math></span> <strong><em>A1</em></strong> <strong><em>N1</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>gradient of normal <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{2}">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math></span> <strong><em>(A1)</em></strong></p>
<p>attempt to substitute their normal gradient and coordinates of P (in any order) <strong><em>(M1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y - 4 = \frac{1}{2}(x - 3),{\text{ }}3 = \frac{1}{2}(4) + b,{\text{ }}b = 1">
<mi>y</mi>
<mo>−</mo>
<mn>4</mn>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>−</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>3</mn>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo stretchy="false">(</mo>
<mn>4</mn>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mi>b</mi>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>b</mi>
<mo>=</mo>
<mn>1</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y - 3 = \frac{1}{2}(x - 4),{\text{ }}y = \frac{1}{2}x + 1,{\text{ }}x - 2y + 2 = 0">
<mi>y</mi>
<mo>−</mo>
<mn>3</mn>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>−</mo>
<mn>4</mn>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>y</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>−</mo>
<mn>2</mn>
<mi>y</mi>
<mo>+</mo>
<mn>2</mn>
<mo>=</mo>
<mn>0</mn>
</math></span> <strong><em>A1</em></strong> <strong><em>N3</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correct answer <strong>and </strong>valid reasoning <strong><em>A2</em></strong> <strong><em>N2</em></strong></p>
<p>answer: <em>eg</em> graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is concave up, concavity is positive (between <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4 < x < 5">
<mn>4</mn>
<mo><</mo>
<mi>x</mi>
<mo><</mo>
<mn>5</mn>
</math></span>)</p>
<p>reason: <em>eg</em> slope of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f’">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
</math></span> is positive, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f’">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
</math></span> is increasing, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f’’ > 0">
<msup>
<mi>f</mi>
<mo>″</mo>
</msup>
<mo>></mo>
<mn>0</mn>
</math></span>,</p>
<p>sign chart (must clearly be for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f’’">
<msup>
<mi>f</mi>
<mo>″</mo>
</msup>
</math></span> and show A and B)</p>
<p><img src="images/Schermafbeelding_2017-08-11_om_10.53.43.png" alt="M17/5/MATME/SP1/ENG/TZ1/06.b/M"></p>
<p> </p>
<p><strong>Note:</strong> The reason given must refer to a specific function/graph. Referring to “the graph” or “it” is not sufficient.</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.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>
<br><hr><br><div class="question">
<p>Consider <em>f</em>(<em>x</em>), <em>g</em>(<em>x</em>) and <em>h</em>(<em>x</em>), for x∈<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathbb{R}">
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span> where <em>h</em>(<em>x</em>) = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {f \circ g} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mi>f</mi>
<mo>∘</mo>
<mi>g</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span>(<em>x</em>).</p>
<p>Given that <em>g</em>(3) = 7 , <em>g′</em> (3) = 4 and <em>f ′ </em>(7) = −5 , find the gradient of the normal to the curve of <em>h</em> at <em>x</em> = 3.</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>recognizing the need to find <em>h′</em> <em><strong> (M1)</strong></em></p>
<p>recognizing the need to find <em>h′ </em>(3) (seen anywhere) <em><strong> (M1)</strong></em></p>
<p>evidence of choosing chain rule <em><strong> (M1)</strong></em></p>
<p><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{{\text{d}}y}}{{{\text{d}}u}} \times \frac{{{\text{d}}u}}{{{\text{d}}x}},\,\,f'\left( {g\left( 3 \right)} \right) \times g'\left( 3 \right),\,\,f'\left( g \right) \times g'">
<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>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>u</mi>
</mrow>
</mfrac>
<mo>×</mo>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>u</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>g</mi>
<mrow>
<mo>(</mo>
<mn>3</mn>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>×</mo>
<msup>
<mi>g</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mn>3</mn>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>g</mi>
<mo>)</mo>
</mrow>
<mo>×</mo>
<msup>
<mi>g</mi>
<mo>′</mo>
</msup>
</math></span></p>
<p>correct working <em><strong>(A1)</strong></em></p>
<p><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( 7 \right) \times 4,\,\, - 5 \times 4">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mn>7</mn>
<mo>)</mo>
</mrow>
<mo>×</mo>
<mn>4</mn>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mo>−</mo>
<mn>5</mn>
<mo>×</mo>
<mn>4</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h'\left( 3 \right) = - 20">
<msup>
<mi>h</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mn>3</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mo>−</mo>
<mn>20</mn>
</math></span> <strong><em>(A1)</em></strong></p>
<p>evidence of taking <strong>their</strong> negative reciprocal for normal <em><strong>(M1)</strong></em></p>
<p><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \frac{1}{{h'\left( 3 \right)}},\,\,{m_1}{m_2} = - 1">
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msup>
<mi>h</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mn>3</mn>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<msub>
<mi>m</mi>
<mn>1</mn>
</msub>
</mrow>
<mrow>
<msub>
<mi>m</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>=</mo>
<mo>−</mo>
<mn>1</mn>
</math></span></p>
<p>gradient of normal is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{{20}}">
<mfrac>
<mn>1</mn>
<mrow>
<mn>20</mn>
</mrow>
</mfrac>
</math></span> <em><strong>A1 N4</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 equation of a curve is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \frac{1}{2}{x^4} - \frac{3}{2}{x^2} + 7">
<mi>y</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mfrac>
<mn>3</mn>
<mn>2</mn>
</mfrac>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>7</mn>
</math></span>.</p>
</div>
<div class="specification">
<p>The gradient of the tangent to the curve at a point P is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 10">
<mo>−<!-- − --></mo>
<mn>10</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>.</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 coordinates of P.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2{x^3} - 3x">
<mn>2</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mi>x</mi>
</math></span> <strong><em>(A1)(A1) (C2)</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="2{x^3}">
<mn>2</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
</math></span>, award <strong><em>(A1) </em></strong>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 3x">
<mo>−</mo>
<mn>3</mn>
<mi>x</mi>
</math></span>.</p>
<p>Award at most <strong><em>(A1)(A0) </em></strong>if there are any extra terms.</p>
<p> </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="2{x^3} - 3x = - 10">
<mn>2</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mi>x</mi>
<mo>=</mo>
<mo>−</mo>
<mn>10</mn>
</math></span> <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(M1) </em></strong>for equating their answer to part (a) to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 10">
<mo>−</mo>
<mn>10</mn>
</math></span>.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = - 2">
<mi>x</mi>
<mo>=</mo>
<mo>−</mo>
<mn>2</mn>
</math></span> <strong><em>(A1)</em>(ft)</strong></p>
<p> </p>
<p><strong>Note: </strong>Follow through from part (a). Award <strong><em>(M0)(A0) </em></strong>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 2">
<mo>−</mo>
<mn>2</mn>
</math></span> seen without working.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \frac{1}{2}{( - 2)^4} - \frac{3}{2}{( - 2)^2} + 7">
<mi>y</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mo stretchy="false">(</mo>
<mo>−</mo>
<mn>2</mn>
<msup>
<mo stretchy="false">)</mo>
<mn>4</mn>
</msup>
</mrow>
<mo>−</mo>
<mfrac>
<mn>3</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mo stretchy="false">(</mo>
<mo>−</mo>
<mn>2</mn>
<msup>
<mo stretchy="false">)</mo>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>7</mn>
</math></span> <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(M1) </em></strong>substituting their <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 2">
<mo>−</mo>
<mn>2</mn>
</math></span> into the original function.</p>
<p> </p>
<p><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> <strong><em>(A1)</em>(ft) <em>(C4)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="( - 2,{\text{ }}9)">
<mo stretchy="false">(</mo>
<mo>−</mo>
<mn>2</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>9</mn>
<mo stretchy="false">)</mo>
</math></span>.</p>
<p> </p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The point A has coordinates (4 , −8) and the point B has coordinates (−2 , 4).</p>
</div>
<div class="specification">
<p>The point D has coordinates (−3 , 1).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the coordinates of C, the midpoint of line segment AB.</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 gradient of the line DC.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of the line DC. Write your answer in the form <em>ax</em> + <em>by</em> + <em>d</em> = 0 where <em>a</em> , <em>b</em> and <em>d</em> are integers.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>(1, −2) <em><strong>(A1)(A1) (C2)</strong></em><br><strong>Note:</strong> Award <em><strong>(A1)</strong></em> for 1 and <em><strong>(A1)</strong></em> for −2, seen as a coordinate pair.</p>
<p>Accept <em>x</em> = 1, <em>y</em> = −2. Award <em><strong>(A1)(A0)</strong></em> if <em>x</em> and <em>y</em> coordinates are reversed.</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="\frac{{1 - \left( { - 2} \right)}}{{ - 3 - 1}}">
<mfrac>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mo>−</mo>
<mn>3</mn>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mfrac>
</math></span> <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for correct substitution, of their part (a), into gradient formula.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - \frac{3}{4}\,\,\,\left( { - 0.75} \right)">
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mn>3</mn>
<mn>4</mn>
</mfrac>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mn>0.75</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong> (C2)</strong></em></p>
<p><strong>Note:</strong> Follow through from part (a).</p>
<p><em><strong>[2 marks]</strong></em></p>
<p> </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="y - 1 = - \frac{3}{4}\left( {x + 3} \right)">
<mi>y</mi>
<mo>−</mo>
<mn>1</mn>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mn>3</mn>
<mn>4</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>3</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>OR </strong></em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y + 2 = - \frac{3}{4}\left( {x - 1} \right)">
<mi>y</mi>
<mo>+</mo>
<mn>2</mn>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mn>3</mn>
<mn>4</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>OR</strong></em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = - \frac{3}{4}x - \frac{5}{4}">
<mi>y</mi>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mn>3</mn>
<mn>4</mn>
</mfrac>
<mi>x</mi>
<mo>−</mo>
<mfrac>
<mn>5</mn>
<mn>4</mn>
</mfrac>
</math></span> <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for correct substitution of their part (b) and a given point.</p>
<p><em><strong>OR</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 = - \frac{3}{4} \times - 3 + c">
<mn>1</mn>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mn>3</mn>
<mn>4</mn>
</mfrac>
<mo>×</mo>
<mo>−</mo>
<mn>3</mn>
<mo>+</mo>
<mi>c</mi>
</math></span> <em><strong>OR</strong></em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 2 = - \frac{3}{4} \times 1 + c">
<mo>−</mo>
<mn>2</mn>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mn>3</mn>
<mn>4</mn>
</mfrac>
<mo>×</mo>
<mn>1</mn>
<mo>+</mo>
<mi>c</mi>
</math></span> <em><strong>(M1) </strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for correct substitution of their part (b) and a given point.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3x + 4y + 5 = 0">
<mn>3</mn>
<mi>x</mi>
<mo>+</mo>
<mn>4</mn>
<mi>y</mi>
<mo>+</mo>
<mn>5</mn>
<mo>=</mo>
<mn>0</mn>
</math></span> (accept any integer multiple, including negative multiples) <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong> (C2)</strong></em></p>
<p><strong>Note:</strong> Follow through from parts (a) and (b). Where the gradient in part (b) is found to be <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{5}{0}">
<mfrac>
<mn>5</mn>
<mn>0</mn>
</mfrac>
</math></span>, award at most <em><strong>(M1)(A0)</strong></em> for either <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = - 3">
<mi>x</mi>
<mo>=</mo>
<mo>−</mo>
<mn>3</mn>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x + 3 = 0">
<mi>x</mi>
<mo>+</mo>
<mn>3</mn>
<mo>=</mo>
<mn>0</mn>
</math></span>.</p>
<p><em><strong>[2 marks]</strong></em></p>
<p> </p>
<p> </p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta ">
<mi>θ<!-- θ --></mi>
</math></span> be an <strong>obtuse</strong> angle such that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\,\theta = \frac{3}{5}">
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ<!-- θ --></mi>
<mo>=</mo>
<mfrac>
<mn>3</mn>
<mn>5</mn>
</mfrac>
</math></span>.</p>
</div>
<div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = {{\text{e}}^x}\,{\text{sin}}\,x - \frac{{3x}}{4}">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mi>x</mi>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mfrac>
<mrow>
<mn>3</mn>
<mi>x</mi>
</mrow>
<mn>4</mn>
</mfrac>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\,\theta "> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</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>Line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="L"> <mi>L</mi> </math></span> passes through the origin and has a gradient of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\,\theta "> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> </math></span>. Find the equation of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="L"> <mi>L</mi> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the derivative of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The following diagram shows the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span> for 0 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> ≤ 3. Line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="M"> <mi>M</mi> </math></span> is a tangent to the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span> at point P.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="M"> <mi>M</mi> </math></span> is parallel to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="L"> <mi>L</mi> </math></span>, find the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-coordinate of P.</p>
<div class="marks">[4]</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>evidence of valid approach <em><strong>(M1)</strong></em></p>
<p><em>eg</em> sketch of triangle with sides 3 and 5, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{co}}{{\text{s}}^2}\,\theta = 1 - {\text{si}}{{\text{n}}^2}\,\theta "> <mrow> <mtext>co</mtext> </mrow> <mrow> <msup> <mrow> <mtext>s</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow> <mtext>si</mtext> </mrow> <mrow> <msup> <mrow> <mtext>n</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> </math></span></p>
<p>correct working <em><strong>(A1)</strong></em></p>
<p><em>eg</em> missing side is 4 (may be seen in sketch), <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,\theta = \frac{4}{5}"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo>=</mo> <mfrac> <mn>4</mn> <mn>5</mn> </mfrac> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,\theta = - \frac{4}{5}"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo>=</mo> <mo>−</mo> <mfrac> <mn>4</mn> <mn>5</mn> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\,\theta = - \frac{3}{4}"> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo>=</mo> <mo>−</mo> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </math></span> <em><strong>A2 N4</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correct substitution of either gradient <strong>or</strong> origin into equation of line <em><strong>(A1)</strong></em></p>
<p>(do not accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = mx + b"> <mi>y</mi> <mo>=</mo> <mi>m</mi> <mi>x</mi> <mo>+</mo> <mi>b</mi> </math></span>)</p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = x\,{\text{tan}}\,\theta "> <mi>y</mi> <mo>=</mo> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y - 0 = m\left( {x - 0} \right)"> <mi>y</mi> <mo>−</mo> <mn>0</mn> <mo>=</mo> <mi>m</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = mx"> <mi>y</mi> <mo>=</mo> <mi>m</mi> <mi>x</mi> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = - \frac{3}{4}x"> <mi>y</mi> <mo>=</mo> <mo>−</mo> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> <mi>x</mi> </math></span> <em><strong>A2 N4</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1A0</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="L = - \frac{3}{4}x"> <mi>L</mi> <mo>=</mo> <mo>−</mo> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> <mi>x</mi> </math></span>.</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}}}{{{\text{d}}x}}\left( {\frac{{ - 3x}}{4}} \right) = - \frac{3}{4}"> <mfrac> <mrow> <mtext>d</mtext> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mo>−</mo> <mn>3</mn> <mi>x</mi> </mrow> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </math></span> (seen anywhere, including answer) <em><strong>A1</strong></em></p>
<p>choosing product rule <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="uv' + vu'"> <mi>u</mi> <msup> <mi>v</mi> <mo>′</mo> </msup> <mo>+</mo> <mi>v</mi> <msup> <mi>u</mi> <mo>′</mo> </msup> </math></span></p>
<p>correct derivatives (must be seen in a correct product rule) <em><strong>A1A1</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,x"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{e}}^x}"> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right) = {{\text{e}}^x}\,{\text{cos}}\,x + {{\text{e}}^x}\,{\text{sin}}\,x - \frac{3}{4}"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>−</mo> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </math></span> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( { = {{\text{e}}^x}\,\left( {{\text{cos}}\,x + {\text{sin}}\,x} \right) - \frac{3}{4}} \right)"> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1 N5</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid approach to equate <strong>their</strong> gradients <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f' = {\text{tan}}\,\theta "> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f' = - \frac{3}{4}"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo>=</mo> <mo>−</mo> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{e}}^x}\,{\text{cos}}\,x + {{\text{e}}^x}\,{\text{sin}}\,x - \frac{3}{4} = - \frac{3}{4}"> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>−</mo> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> <mo>=</mo> <mo>−</mo> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{e}}^x}\,\left( {{\text{cos}}\,x + {\text{sin}}\,x} \right) - \frac{3}{4} = - \frac{3}{4}"> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> <mo>=</mo> <mo>−</mo> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </math></span></p>
<p>correct equation without <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{e}}^x}"> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </math></span> <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\,x = - {\text{cos}}\,x"> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,x + {\text{sin}}\,x = 0"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>=</mo> <mn>0</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{ - {\text{sin}}\,x}}{{{\text{cos}}\,x}} = 1"> <mfrac> <mrow> <mo>−</mo> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mn>1</mn> </math></span></p>
<p>correct working <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\,\theta = - 1"> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 135^\circ "> <mi>x</mi> <mo>=</mo> <msup> <mn>135</mn> <mo>∘</mo> </msup> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{{3\pi }}{4}"> <mi>x</mi> <mo>=</mo> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> </math></span> (do not accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="135^\circ "> <msup> <mn>135</mn> <mo>∘</mo> </msup> </math></span>) <em><strong>A1 N1</strong></em></p>
<p><strong>Note:</strong> Do not award the final <em><strong>A1</strong></em> if additional answers are given.</p>
<p><em><strong>[4 marks]</strong></em></p>
<p> </p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mfrac><mn>3</mn><mi>x</mi></mfrac><mo>,</mo><mo> </mo><mi>x</mi><mo>≠</mo><mn>0</mn></math>.</p>
</div>
<div class="specification">
<p>Line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> is a tangent to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>,</mo><mo> </mo><mo>−</mo><mn>2</mn><mo>)</mo></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced></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>Use your answer to part (a) to find the gradient of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</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>Determine the number of lines parallel to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> that are tangent to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>. Justify your answer.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo></mrow></mfenced><mo> </mo><mn>2</mn><mi>x</mi><mo>+</mo><mfrac><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></mfrac></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>2</mn><mi>x</mi></math>, <em><strong>A1</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>+</mo><mfrac><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></mfrac></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>3</mn><msup><mi>x</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></math><br> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to substitute <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> into their part (a) <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>f</mi><mo>'</mo><mfenced><mn>1</mn></mfenced><mo>=</mo></mrow></mfenced><mo> </mo><mn>2</mn><mfenced><mn>1</mn></mfenced><mo>+</mo><mfrac><mn>3</mn><msup><mn>1</mn><mn>2</mn></msup></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math> <em><strong>A1</strong></em><br> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>=</mo><mn>2</mn><mi>x</mi><mo>+</mo><mfrac><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>686</mn><mo>,</mo><mo> </mo><mn>1</mn><mo>,</mo><mo> </mo><mn>2</mn><mo>.</mo><mn>19</mn><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mo>-</mo><mn>0</mn><mo>.</mo><mn>686140</mn><mo>…</mo><mo>,</mo><mo> </mo><mn>1</mn><mo>,</mo><mo> </mo><mn>2</mn><mo>.</mo><mn>18614</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><strong>OR</strong></p>
<p>sketch of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced></math> with line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>5</mn></math> <em><strong>M1</strong></em></p>
<p><img src="data:image/png;base64,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"></p>
<p>three points of intersection marked on this graph <em><strong>A1</strong></em></p>
<p>(and it can be assumed no further intersections occur outside of this window)</p>
<p><br><strong>THEN</strong></p>
<p>there are two other tangent lines to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> that are parallel to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> The final <em><strong>A1</strong> </em>can be awarded provided two solutions other than <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn></math> are shown <strong>OR</strong> three points of intersection are marked on the graph.</p>
<p>Award <em><strong>M1A1A1</strong></em> for an answer of “3 lines” where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> is considered to be parallel with itself (given guide definition of parallel lines), but only if working is shown.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Was reasonably well done, with the stronger candidates able to handle a negative exponent appropriately when finding the derivative. There were a few who confused the notation for derivative with the notation for inverse.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Most knew to substitute <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn></math> into the derivative to find the gradient at that point, but some also tried to substitute the <em>y</em>-coordinate for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo></math>.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>There was a lot of difficulty understanding what approach would help them determine the number of tangents to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> that are parallel to <em>L</em>. Several wrote just an answer, which is not adequate when justification is required.</p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p>The graph of a function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> passes through the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>ln</mi><mo> </mo><mn>4</mn><mo>,</mo><mo> </mo><mn>20</mn></mrow></mfenced></math>.</p>
<p>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>6</mn><msup><mtext>e</mtext><mrow><mn>2</mn><mi>x</mi></mrow></msup></math>, find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced></math>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>evidence of integration <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo> </mo><mtext>d</mtext><mi>x</mi><mo> </mo><mo>,</mo><mo> </mo><mo>∫</mo><mn>6</mn><msup><mtext>e</mtext><mrow><mn>2</mn><mi>x</mi></mrow></msup></math></p>
<p>correct integration (accept missing <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>+</mo><mi>c</mi></math>) <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mn>6</mn><msup><mtext>e</mtext><mrow><mn>2</mn><mi>x</mi></mrow></msup><mo> </mo><mo>,</mo><mo> </mo><mn>3</mn><msup><mtext>e</mtext><mrow><mn>2</mn><mi>x</mi></mrow></msup><mo>+</mo><mi>c</mi></math></p>
<p>substituting initial condition into <strong>their</strong> integrated expression (must have <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>+</mo><mi>c</mi></math>) <em><strong>M1</strong></em></p>
<p><em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><msup><mtext>e</mtext><mrow><mn>2</mn><mo>×</mo><mi>ln</mi><mo> </mo><mn>4</mn></mrow></msup><mo>+</mo><mi>c</mi><mo>=</mo><mn>20</mn></math></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>M0</strong></em> if candidate has substituted into <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>''</mo></math>.</p>
<p> </p>
<p>correct application of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>log</mi><mfenced><msup><mi>a</mi><mi>b</mi></msup></mfenced><mo>=</mo><mi>b</mi><mo> </mo><mi>log</mi><mo> </mo><mi>a</mi></math> rule (seen anywhere) <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><mi>ln</mi><mo> </mo><mn>4</mn><mo>=</mo><mi>ln</mi><mo> </mo><mn>16</mn><mo> </mo><mo>,</mo><mo> </mo><msup><mtext>e</mtext><mrow><mi>ln</mi><mo> </mo><mn>16</mn></mrow></msup><mo> </mo><mo>,</mo><mo> </mo><mi>ln</mi><mo> </mo><msup><mn>4</mn><mn>2</mn></msup></math></p>
<p>correct application of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mi>ln</mi><mo> </mo><mi>a</mi></mrow></msup><mtext>=</mtext><mtext mathvariant="italic">a</mtext></math> rule (seen anywhere) <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mi>ln</mi><mo> </mo><mn>16</mn></mrow></msup><mo>=</mo><mn>16</mn><mo> </mo><mo>,</mo><mo> </mo><msup><mfenced><msup><mtext>e</mtext><mrow><mi>ln</mi><mo> </mo><mn>4</mn></mrow></msup></mfenced><mn>2</mn></msup><mo>=</mo><msup><mn>4</mn><mn>2</mn></msup></math></p>
<p>correct working <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>×</mo><mn>16</mn><mo>+</mo><mi>c</mi><mo>=</mo><mn>20</mn><mo> </mo><mo>,</mo><mo> </mo><mn>3</mn><mo>×</mo><msup><mfenced><mn>4</mn></mfenced><mn>2</mn></msup><mo>+</mo><mi>c</mi><mo>=</mo><mn>20</mn><mo> </mo><mo>,</mo><mo> </mo><mi>c</mi><mo>=</mo><mo>-</mo><mn>28</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>3</mn><msup><mtext>e</mtext><mrow><mn>2</mn><mi>x</mi></mrow></msup><mo>-</mo><mn>28</mn></math> <em><strong>A1 N4</strong></em></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="question">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f’(x) = \frac{{3{x^2}}}{{{{({x^3} + 1)}^5}}}">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mrow>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mrow>
<msup>
<mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
<mn>5</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span>. Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(0) = 1">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>1</mn>
</math></span>, find <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>.</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>valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {f'{\text{d}}x,{\text{ }}\int {\frac{{3{x^2}}}{{{{({x^3} + 1)}^5}}}{\text{d}}x} } ">
<mo>∫</mo>
<mrow>
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo>∫</mo>
<mrow>
<mfrac>
<mrow>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mrow>
<msup>
<mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
<mn>5</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mrow>
</math></span></p>
<p>correct integration by substitution/inspection <strong><em>A2</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = - \frac{1}{4}{({x^3} + 1)^{ - 4}} + c,{\text{ }}\frac{{ - 1}}{{4{{({x^3} + 1)}^4}}}">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
<mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
<msup>
<mo stretchy="false">)</mo>
<mrow>
<mo>−</mo>
<mn>4</mn>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mi>c</mi>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mfrac>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mrow>
<mn>4</mn>
<mrow>
<msup>
<mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
<mn>4</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span></p>
<p>correct substitution into <strong>their </strong>integrated function (must include <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c">
<mi>c</mi>
</math></span>) <strong><em>M1</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 = \frac{{ - 1}}{{4{{({0^3} + 1)}^4}}} + c,{\text{ }} - \frac{1}{4} + c = 1">
<mn>1</mn>
<mo>=</mo>
<mfrac>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mrow>
<mn>4</mn>
<mrow>
<msup>
<mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mn>0</mn>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
<mn>4</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mo>+</mo>
<mi>c</mi>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
<mo>+</mo>
<mi>c</mi>
<mo>=</mo>
<mn>1</mn>
</math></span></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>M0 </em></strong>if candidates substitute into <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f’">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f’’">
<msup>
<mi>f</mi>
<mo>″</mo>
</msup>
</math></span>.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c = \frac{5}{4}">
<mi>c</mi>
<mo>=</mo>
<mfrac>
<mn>5</mn>
<mn>4</mn>
</mfrac>
</math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = - \frac{1}{4}{({x^3} + 1)^{ - 4}} + \frac{5}{4}{\text{ }}\left( { = \frac{{ - 1}}{{4{{({x^3} + 1)}^4}}} + \frac{5}{4},{\text{ }}\frac{{5{{({x^3} + 1)}^4} - 1}}{{4{{({x^3} + 1)}^4}}}} \right)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
<mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
<msup>
<mo stretchy="false">)</mo>
<mrow>
<mo>−</mo>
<mn>4</mn>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mfrac>
<mn>5</mn>
<mn>4</mn>
</mfrac>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mrow>
<mn>4</mn>
<mrow>
<msup>
<mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
<mn>4</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mn>5</mn>
<mn>4</mn>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mfrac>
<mrow>
<mn>5</mn>
<mrow>
<msup>
<mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
<mn>4</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mrow>
<mn>4</mn>
<mrow>
<msup>
<mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
<mn>4</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>A1</em></strong> <strong><em>N4</em></strong></p>
<p><strong><em>[6 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = \frac{1}{{\sqrt {2x - 1} }}">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</msqrt>
</mrow>
</mfrac>
</math></span>, for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x > \frac{1}{2}">
<mi>x</mi>
<mo>></mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</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="\int {{{\left( {f\left( x \right)} \right)}^2}{\text{d}}x} ">
<mo>∫</mo>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Part of the graph of <em>f</em> is shown in the following diagram.</p>
<p><img src="data:image/png;base64,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"></p>
<p>The shaded region <em>R</em> is enclosed by the graph of <em>f</em>, the <em>x</em>-axis, and the lines <em>x</em> = 1 and <em>x</em> = 9 . Find the volume of the solid formed when <em>R</em> is revolved 360° about the <em>x</em>-axis.</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>correct working <em><strong>(A1)</strong></em></p>
<p>eg <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {\frac{1}{{2x - 1}}{\text{d}}x,\,\,\int {{{\left( {2x - 1} \right)}^{ - 1}},\,\,\frac{1}{{2x - 1}},\,\,\int {{{\left( {\frac{1}{{\sqrt u }}} \right)}^2}\frac{{{\text{d}}u}}{2}} } } ">
<mo>∫</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mo>∫</mo>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mfrac>
<mn>1</mn>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mo>∫</mo>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mi>u</mi>
</msqrt>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>u</mi>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
</mrow>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\int {\left( {f\left( x \right)} \right)} ^2}{\text{d}}x = \frac{1}{2}{\text{ln}}\left( {2x - 1} \right) + c">
<mrow>
<mo>∫</mo>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mtext>ln</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>c</mi>
</math></span> <em><strong>A2 N3</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="\frac{1}{2}{\text{ln}}\left( {2x - 1} \right)">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mtext>ln</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to substitute either limits or the function into formula involving <em>f </em><sup>2</sup> (accept absence of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi ">
<mi>π</mi>
</math></span> / d<em>x</em>) <strong><em>(M1)</em></strong></p>
<p><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_1^9 {{y^2}{\text{d}}x,\,\,} \pi {\int {\left( {\frac{1}{{\sqrt {2x - 1} }}} \right)} ^2}{\text{d}}x,\,\,\left[ {\frac{1}{2}{\text{ln}}\left( {2x - 1} \right)} \right]_1^9">
<msubsup>
<mo>∫</mo>
<mn>1</mn>
<mn>9</mn>
</msubsup>
<mrow>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</mrow>
<mi>π</mi>
<mrow>
<mo>∫</mo>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
</msqrt>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<msubsup>
<mrow>
<mo>[</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mtext>ln</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>]</mo>
</mrow>
<mn>1</mn>
<mn>9</mn>
</msubsup>
</math></span></p>
<p>substituting limits into <strong>their</strong> integral and subtracting (in any order) <strong><em>(M1)</em></strong></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{\pi }{2}\left( {{\text{ln}}\left( {17} \right) - {\text{ln}}\left( 1 \right)} \right),\,\,\pi \left( {0 - \frac{1}{2}{\text{ln}}\left( {2 \times 9 - 1} \right)} \right)">
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>17</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>π</mi>
<mrow>
<mo>(</mo>
<mrow>
<mn>0</mn>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mtext>ln</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mo>×</mo>
<mn>9</mn>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p>correct working involving calculating a log value or using log law <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\left( 1 \right) = 0,\,\,{\text{ln}}\left( {\frac{{17}}{1}} \right)">
<mrow>
<mtext>ln</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ln</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>17</mn>
</mrow>
<mn>1</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{\pi }{2}{\text{ln}}17\,\,\,\,\left( {{\text{accept }}\pi {\text{ln}}\sqrt {17} } \right)">
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
<mrow>
<mtext>ln</mtext>
</mrow>
<mn>17</mn>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>accept </mtext>
</mrow>
<mi>π</mi>
<mrow>
<mtext>ln</mtext>
</mrow>
<msqrt>
<mn>17</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1 N3</strong></em></p>
<p><strong>Note:</strong> Full <em><strong>FT</strong></em> may be awarded as normal, from their incorrect answer in part (a), however, do not award the final two <em><strong>A</strong></em> marks unless they involve logarithms.</p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mn>2</mn><mi>x</mi></mfrac><mo>+</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>3</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>≠</mo><mn>0</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of the normal to the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>1</mn><mo>,</mo><mo> </mo><mn>2</mn></mrow></mfenced></math> in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi><mi>y</mi><mo>+</mo><mi>d</mi><mo>=</mo><mn>0</mn></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>,</mo><mo> </mo><mi>d</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>-</mo><mn>2</mn><msup><mi>x</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup><mo>+</mo><mn>6</mn><mi>x</mi></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>-</mo><mfrac><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></mfrac><mo>+</mo><mn>6</mn><mi>x</mi></math> <em><strong>A1(M1)A1</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>6</mn><mi>x</mi></math> seen, and <em><strong>(M1)</strong></em> for expressing <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mi>x</mi></mfrac></math> as <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> (this can be implied from either <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></mfrac></math> seen in their final answer), <em><strong>A1</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></mfrac></math>. Award at most <em><strong>A1(M1)A0</strong></em> if any additional terms are seen.</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>finding gradient at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><menclose notation="right"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></menclose><mrow><mi>x</mi><mo>=</mo><mn>1</mn></mrow></msub><mo>=</mo><mn>4</mn></math> <em><strong>A1</strong></em></p>
<p>finding the perpendicular gradient <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>m</mi><mo>⊥</mo></msub><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mfenced><mn>1</mn></mfenced><mo>+</mo><mi>c</mi></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>-</mo><mn>2</mn><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>M1</strong> </em>for correctly substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>2</mn></math> and their <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>m</mi><mo>⊥</mo></msub></math>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>+</mo><mn>4</mn><mi>y</mi><mo>-</mo><mn>9</mn><mo>=</mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Do not award the final <em><strong>A1</strong> </em>if the answer is not in the required form. Accept integer multiples of the equation.</p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Differentiating the function was challenging for many candidates. The most frequently obtained mark was for the term <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mi>x</mi></math>. Handling the <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><mi>x</mi></mfrac></math> term was problematic and consequently the method mark and final accuracy mark were lost.</p>
<p> </p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Some good attempts at finding the equation of the normal were seen amongst the few that answered this part. Of those that found an equation in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi><mi>y</mi><mo>+</mo><mi>d</mi><mo>=</mo><mn>0</mn></math> most included fractions thus hardly any fully correct answers were seen.</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The diagram shows the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><msup><mi>x</mi><mn>2</mn></msup><mn>2</mn></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn><mi>a</mi></mrow><mi>x</mi></mfrac><mo>,</mo><mo> </mo><mi>x</mi><mo>≠</mo><mn>0</mn></math>.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAASQAAADzCAYAAAAvtWvPAAAWrklEQVR4Ae2dvWscSRqH599Q7PSyZXNFTi7wJZtc4MAYNtjYAmeXmAsEhsMLFwnO2SUDm+6CODicLAg2uMAoO85GybEcxrDLYkQdP0lljUYz3dXd9fFW1dMgNNNdXfXW73376frqnpVjQwEUQAEjCqyM2IEZKIACKOAAEkGAAihgRgGAZMYVGIICKACQiAEUQAEzCgAkM67AEBRAAYBEDKAACphRACCZcQWGoAAKACRiAAVQwIwCAMmMKzAEBVAAIBEDKIACZhQASGZcgSEogAIAiRhAARQwowBAMuMKDEEBFABIXcbAJ3ex/tqtViu3Wv3eHZ/971qFD6fu6OArd3L+a5eqUOnyCgCk8j4oZ8EVgA7cw5O37lJWXL53p8+fuOOzj+VsouSuFQBIPbv/05k7frByD47P3CfpcPlvt37+V3f28QpPPStD3QspAJAKCW+j2P+49eMH7uDo1H1wl+7D6bfupe++2TAQKzpTACB15vC71f2vOz36wq0er937i1P38uTM0Vm7qxDf8ioAkPLqbay0j+7s+NCtDr9xR0d01Yw5p0tzAFKXbveV/tWdn3zlVgffuPX73/xO/qNAMQUAUjHpLRT8qzv/25/dydsPFozBBhTgndrdxYBm0p48dEfrf7ofXh67v/x4cT3lfyPETz/9dLU+6fz8vDtpqHB5BWghlfdBXgsu37qThwfu4PG37seLu920X375xX355ZdXQHr06JHTdzYUyKkAQMqptvGyXr9+fbN6Wyu4V07f2VAgpwIAKafahstSF00QWq/Xd/6/e/fOsNWY1poCAKk1j86oj7pm6qI9ffr06myBaXvfjGw5BQUmKwCQJkvW3gm+VeRbQwKSts1WU3u1pkYWFQBIFr2S0SZBaHu8yANJZrx69epqoPvnn3/OaBVF9aoAQOrV8zf1Vjdte0ZtE0h+5u3Zs2edK0X1cygAkHKobLQM3zraXnO0CSSZ7tcmGa0GZjWkAEBqyJmxqrINpFj5kg8KjCkAkMYU6vA4QOrQ6UaqDJCMOMKSGQDJkjf6sgUg9eXvoNoCpCCZSJRAAYCUQNTaswRItXuwXvsBUr2+S2b5EJCY/k8mu6mMte7sxYsX2W0CSNklt1/gEJCGjtmvGRaGKqAFsfrLvQGk3IpXUN4QdIaOVVA1TAxQQK0j+bnE6nyAFOCg3pIMQWfoWG86tVpftYxKvXoGILUaVQvqNQSdoWMLiuRUIwpo9b5e0lfq5XzJgKTHEUo0+Yz4tWozhqAzdKzqSmP8lQKatNDbH0ptyYCkSpUYFCslZEvlDkFn6FhLGvRYFzUiSraOpHkyIPmnxGkl1RfaQ9AZOlZfTbF4UwG9+eHNmzebu7J/TgYk1YRWUnZ/RilwCDpDx6IUTiZFFPj+++8/vzG0iAE3hSYFkm8l+TcRlqwoZYcrMASdoWPhJZDSkgL+Ot1+DU0JG5MCSRVSK4nVvSVcO7/MIejIl9xg5mtr8UxN8ZdYlb1Li+RAskTfXQKw774CY0CycCe9bzV75ijgp/mtjPUmB5JEopU0J1TKnQOQymmfu2QNZOv6tLJlARKtJCvuDrMDIIXpVHsqKwPZmzpmAZIK1HSi/92vTQP4bE8BgGTPJ7EtUhdNa46sdb+zAUmC6tctSq9ziO3YFvMDSC169W6dNDlR6nm1u5bc/ZYVSIKRoMRmWwGAZNs/S63Tr8hs//TV0jxjnZ8VSDK69LMysYRrOZ8hIGkA1NIgaMt+SFE36+O52YFk4XmZFI5uKU+A1JI379ZF640sdtW8ldmBpIKti+LF6fU/QGrT85pVs9pV84oXAZJG+BX0VhZjeTH4f60AQGovEqzOqm0rXQRIMsLScvVtUXr/PgQkTUxYecygdz9Nqb/VWbXtOhQDkvXBtW2hevo+BCSNASq42epRQJMQtawBLAYkudPiStF6wiydpQApnba5c/bPqtXyQHRRIMk5LJbMHaLj5Y0BqZa77XhN206hXoiuL934a9mKA4llAPZCZQhIsnbsuL0a9WmRxvpqG+8rDiSFCoslbV0wY8AZO26rNn1aU8MU/y7PmAASywB2uabcvjHg6Li6A2w2FdB4kXxUy7jRpoomgCSDtAyA2ZtN15T7PAYk+cnaU+Ll1LJVco3jRpsKmgESywA23VL2M0Aqq/+S0mscN9qsrxkgySjeBrDpmnKfx4CkoOc1MuX8s69kv96o5u60KSBJaAa494Vbvv1jQFLg64/NjgJ6pYheuFbjuNGmiuaA5JcB8Jzbppvyfg4BkuUnxvOqVb40QUgwEpRq38wBSYLqJ7hrWz9ReyBs2j8GJB4f2VSr7Gd1z6y9qH+JIiaBxAD3EpcuPxcgLdcwVw61D2Jv62QSSDLSL+zaNpjv6RUYA5JuGGNp0ltJCeo2q3VU8yD2thfNAkmGMsC97a4830NgE5Imj7V9lqJZTo0btTbWahpIfrCuNdGtX0IhsNGdmcWRZTwp3eWj2mfUdqlnGkgymBXcu9yWdl8IkNR6bWFWJ62S8XP3N+lW14GZB5L6x3qFAsEfP7j35RgCJNYi7VMv3X5/LbS8Bsw8kORev+irpcG7dGG7POcQIGnSQcsz2PIooNhXN7l1zasAklyu6U0W4+UJ/hAgsRYpjy98KYp/dZNb36oBkga2Lf4WeYsBEgIk+SMkXYv65K6TWkWtTe/v07AaIKkCrE3a58a4+0NBo3TMgMbVfju3nmCkulcFJBnM2qTtkI3/PRRI8gVT//H19zn6tUYtTu/7Om7/rw5Ico4umJ6ctO201N9DgaS7d8szPql1Hsq/RxhJj+qAJKN1EahPzZZGgVAgqQvNQ9DxfdArjKRklUCS4QISd+f4F8NVUKzCwkLdNW4McX0gGPXcAwiLvLiaR8nNr1hlUDWKnHcyCW0h6aQpae8Uwpd7CngY6X+vW7VAksPUQuphbUbu4JwCGbWQGNhe7iE/NtozjKRi1UBSBXRBaCyDLZ4CU4DEwPZy3X1rv3cYScnqgeSdSddt+YXhc5gCJAa2vWrz/qt1qQW/wOhav+qBpGow6zbvYth31hQg+RvCvrzYv18BQUhaA6NbjZoAkqrDrNutU5d+mgIklaU7vMDEFq4AMNqtVTNA0gWhC4kLY7ejp+ydCiStRWIcL1xhYLRfq2aApCrSddvv6ClHpgKJcaRwdQUjWpT79WoKSKqmum68pmS/w0OOTAUS40ghql4/HA6MhrVqDkiabZPTWRsz7Piho1OBpLz0Vk+6y/tV1fIIYLRfH3+kOSCpYupC6ALhDZPezdP+zwES65H2ayxt1HJnacp+jfyRJoGkymmglQc/vZun/Z8DJL1mWBcd260C/rWz0oWb460uQ5+aBZICQE1k1ngMuX/3sTlAUk46j1bAtabqvgpEah0Bo91xtmtvs0BSZf0qWC6SXa7fv28ukJj+v9bUD/ILRmzTFGgaSJLC/9zwNFn6Tj0XSGqN9t5tY43RsmuneSBJHl0kLAUID5S5QPLd5F5bpIoxZnjD42xXyi6ApAtEgcKPTe4Kgfv75gJJOamb0hv8BWK9Bkc3PpY+3I+nKXu6AJIE4ccmw8NiCZD8uF14aXWnFIC0xETjZwxeL/dlN0CSVLp780K38aBZAiTlrgu0h9lN1VEtb16lPB5ToSm6ApLuYIwnjYfGUiDpQm0Z/Ioj3dwYLxqPpakpugKSxGE8aTxElgJJF2yrF6tfXyTg0kUbj6WpKboDkgTy40m9zgaNBclSICn/Ft93rkeSpA1dtLEImn+8SyBJLr8+ibvc/eCJAaSWWkm6calFxAPE92Ml9p5ugSQhFWSspr0fUjGApFxbaCWpVaTup25g3Lzux0rsPV0DSQGmu56Cju1WgVhAUo61zrhttop4lc1tbKT+1DWQJK5/7ohFk7ehFhNIfryuptYFraLbWMj9qXsgSXC/noRVttfhFxNIyrGWV8GoJaRlIfqjVZQbRTexV6ZYe6VqvEOBWNOdPJWKsYEkTTUOY3WxpLpnfl0R3fdUURWWLy2kDZ0UlIJS71tsIElP/0iJpVaoQKkbkeor33MzKh/5AGnDBwpIAan3mbcUQJLMvmtc+sL3IFKrTd1J1qNtXASFPwKkLQcoWDUz1PPit1RAktS+FVoCSpsg0pIPxom2gt/AV4C0wwl+5s3qmMcOk6PuSgkkGSooqXWSq/umclSm75oBoqjhEjUzgLRHTgWxArjH4E0NJEnuu2+pBpHVGlIZ6oL7hY10zfYEu6HdAGnAGf6iyXUnHzAl66EcQFKFpKu6x+o+xYCFh5DGhVQH/e+1lZs1YCIWBpBGxOwRSrmA5KXXeJ0fYJ6yQFUAUnq/ZMNDSK2uGIDz9vE/nwIAKUDr3h7EzQ0kucC3btRaUvn+OUPBZvNPrR4dE8D0p886LjApD7a6FQBIgf4rOTsUaGK0ZCWAtGm8wKKxO7V0NmHkwaNjtIA2FWvnM0Ca4MteoFQaSBNcQtLGFABIEx2qLkLrq7kB0sSgIHk0BQDSRCnVnWh9NTdAmhgUJI+mAECaIWXrUAJIM4KCU6IoAJBmytgylADSzKDgtMUKAKQFErYKJYC0ICg4dZECAGmRfNfrZ1obUwJIC4OC02crAJBmS3d7YmstJYB061s+5VUAIEXSuyUoAaRIQUE2kxUASJMl239CK1ACSPt9zJG0CgCkyPq2ACWAFDkoyC5YAYAULFV4wtqhBJDCfd1LSj07qIfMUz9DCJASRZSHkp5O1+eaNoBUk7fy2CoQ6e0Kig095JxqA0iplL15pYaWBOivJigBpIRBUXHWimG1khQfiukUb1MFSIkDRE5UK6kmKAGkxEFRefYCkd70qTgRoGLebAFSpuCo6dUlAClTUFRejLpuihXBacqbPoeqDZCG1Il8TFCS86y/oxsgRXZ8w9kplvVKHt9aWlrVSUBSofyhATFADOyKAXXflm6TgLS0MM6/VsD6Dwco2NhQYEwBumxjClV0XFDShW/xZ3oAUkWBVMBUBrULiJ6jSA0Eam2HNSgBpBzer68Mpv3r89lkizUomBVKH9+69dHD67HAw+duff7hns0A6Z4k3e8IWxj5H7d+/OAqtg6erN37y9/cxY/fuscHK7d6vHYXASoyWBAgUuok2aB0+W+3fvI79zlYTv/kDg++cev3v92pIkC6IwdfnLt6ZETr6cYfHbl0H89eusPVH9zxybE7OvmX+zhBQYA0QayUSdNDyQfKV+7k/Nfrqly+dScPD9zB0anbbCcBpJSe7iBvH1dXraRp9QVI0/RKmtpDKdYis7vGfnRnx4du9eDYnX3yR3btc1dNbp+C/ygwXYHruNq+0YXkA5BCVMqYJt2SgJv+/cMTd37pK/TJXay/dqvV12598ZlSAMnLw/9ZCly+/849f/JHd3jw3J1++BxsQXkBpCCZ8ibST0hrRXfMZ4ScuwHSncFFgJTXsx2U9vFf7uTor+7s/Q/u6ODQHZ/97N5/93f3j0AwASSjMaLHTLQkP94GkOJpSU73FLgZN1p9nrn9rzs9+sKtDh67lz9euNB2EkC6p6yNHWod6Q0B8d49s2u8aNc+xpBsRECfVgAkw373g9xx3jtz6T6cPncHK2bZDLu8e9MAkvEQiDqedLUO6Qt3+PwHd3H5wZ2vn7MOybj/ezMPIFXgcY0lxXiSWlW9vHjjXj7+HSu1K/B7jyYCpAq87pftx+m6jVeYhZHjGpEijQIAKY2u0XP1XbfoGe/IECDtEIVdWRQASFlkjlNI3Fm3/TYBpP3acCStAgAprb5Rc1eXTW8GiLtg8r6JAOm+JuzJowBAyqNztFK0YFJ/KTeAlFJd8h5SACANqWPwmAa4BYzx10DMNx4gzdeOM5cpAJCW6VfkbK3eTtlKAkhF3EqhzjmAVGEYaAxJY0mpWkkAqcKgaMRkgFSpI1O2kgBSpUHRgNkAqVInpmwlAaRKg6IBswFSxU5MNeMGkCoOispNB0gVO9DPuMVelwSQKg6Kyk0HSJU7UL8EEe+dSddiAKTKg6Ji8wFSxc6T6f5XRGNWAyDFVJO8pigAkKaoZTSt3r8d85dKAJJRR3dgFkBqwMl6E4C6brE2gBRLSfKZqgBAmqqYwfQa1BZEYg1uAySDTu7EJIDUiKPVQlJLKcYGkGKoSB5zFABIc1QzeI5+YFLvS4qxAaQYKpLHHAUA0hzVjJ4T6/k2gGTUwR2YBZAacrJWbsfotgGkhoKisqoApMocNmSupv5jdNsA0pDKHEupAEBKqW6BvGN02wBSAcdR5JUCAKmxQIgx2waQGguKiqoDkCpyVoipmm1bukgSIIUoTZoUCgCkFKoWzDPGIkmAVNCBnRcNkBoMAA1sL3m2DSA1GBSVVAkgVeKoKWbqdSSvX7+ecsqdtADpjhx8yagAQMoodq6ilr6SBCDl8hTlbCsAkLYVaeS7oDL3V0kAUiNBUGE1AFKFTgsx+dmzZ04zbnM2gDRHNc6JoQBAiqGiwTyW/EwSQDLo0E5MAkiNOlrjSHMfIwFIjQZFBdUCSBU4aa6JAsucl7YBpLmKc95SBQDSUgUNn69xJLWUpm4AaapipI+lAECKpaTBfLQWac5PJAEkg87sxCSA1LCjNcumVtLUDSBNVYz0sRQASLGUNJiP1iHpdSRTN4A0VTHSx1IAIMVS0mg+gsvUBZIAyagzOzALIDXu5DkD2wCp8aAwXD2AZNg5MUybM7ANkGIoTx5zFABIc1Sr6Jw5v2oLkCpycGOmAqTGHLpdnTkrtgHStop8z6UAQMqldMFypgJmavqCVaPoxhQASI05dFd1NPX/7t27XYd27gNIO2VhZwYFAFIGkUsXMXWmDSCV9li/5QOkDnw/daYNIHUQFEarCJCMOiamWVPfsQ2QYqpPXlMUAEhT1Ko0rWbapjzTBpAqdXQDZgOkBpw4VoWpU/8AaUxRjqdSACClUtZYvlMgMyWtsWpiTuUKAKTKHRhqviAT+vZIgBSqKuliKwCQYitqNL8pU/8AyagTOzALIHXgZFURIHXi6MqrCZAqd2Co+VPWItFCClWVdLEVAEixFTWan9Yihb5fGyAZdWIHZgGkDpysKgpGr169CqotQAqSiUQJFABICUS1mOWUxZEAyaIH+7AJIPXh56vfZwtdrQ2QOgkKg9UESAadksIkvX7k0aNHQVkDpCCZSJRAAYCUQFSrWYaCJjSd1XpiV70KAKR6fTfZ8lDQhKabbAAnoMCIAgBpRKCWDoeCJjRdS9pQFxsKACQbfshixdOnT68Gt8cKA0hjCnE8lQIAKZWyBvMNfXwEIBl0XicmAaROHK1qAqSOnF1pVQFSpY6bYzZAmqMa5+RUACDlVLtwWS9evHBv3rwZtYIu26hEJEikAEBKJKzFbEMfsAVIFr3Xh00AqQ8/X9USIHXk7EqrCpAqddwcswHSHNU4J6cCACmn2oXLAkiFHUDxowoApFGJ2kmgV5Dob2xjDGlMIY6nUgAgpVK24nwBUsXOq9x0gFS5A1OYD5BSqEqeIQoApBCVOksDkDpzuKHqAiRDzrBiCkCy4on+7ABI/fl8tMYAaVQiEiRSACAlErbmbAFSzd6r23aAVLf/klgPkJLISqYBCgCkAJF6SwKQevO4nfoCJDu+MGMJQDLjiu4MAUjduXy8wgBpXCNSpFEAIKXRtepcAVLV7qvaeIBUtfvSGA+Q0uhKruMKAKRxjbpLAZC6c7mZCgMkM67AEBRAAYBEDKAACphRACCZcQWGoAAKACRiAAVQwIwC/wfCE63Hag32BAAAAABJRU5ErkJggg=="></p>
<p>The equation of the vertical asymptote of the curve is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>k</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mtext>d</mtext><mi>y</mi></mrow><mrow><mtext>d</mtext><mi>x</mi></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>At the point where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>2</mn></math>, the gradient of the tangent to the curve is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>5</mn></math>.</p>
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>k</mi><mo>=</mo></mrow></mfenced><mo> </mo><mn>0</mn></math> <em><strong>(A1)</strong></em><em><strong> (C1)</strong></em> </p>
<p><strong>Note:</strong> Award <em><strong>(A1)</strong></em> for an answer of "<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math>".</p>
<p><em><strong><span class="mjpage">[1 mark]</span></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>x</mi><mo>-</mo><mfrac><mrow><mn>2</mn><mi>a</mi></mrow><msup><mi>x</mi><mn>2</mn></msup></mfrac></math> <em><strong>(A1)(A1)(A1)</strong></em><em><strong> (C3)</strong></em> </p>
<p><strong>Note:</strong> Award <em><strong>(A1)</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>, <em><strong>(A1)</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><mi>a</mi></math>, <em><strong>(A1)</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><msup><mi>x</mi><mn>2</mn></msup></mfrac></math>. Award at most <em><strong>(A1)(A1)(A0)</strong></em> if extra terms are seen.</p>
<p><em><strong><span class="mjpage">[3 marks]</span></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"><mn>0</mn><mo>.</mo><mn>5</mn><mo>=</mo><mn>2</mn><mo>-</mo><mfrac><mrow><mn>2</mn><mi>a</mi></mrow><msup><mn>2</mn><mn>2</mn></msup></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for <em>their</em> correctly substituted derivative equated to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>5</mn></math>.</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>a</mi><mo>=</mo></mrow></mfenced><mo> </mo><mn>3</mn></math> <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong> (C2)</strong></em></p>
<p><strong>Note:</strong> Follow through from part (b) providing their answer is <strong>not</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>0</mn></math> as this value contradicts the graph.</p>
<p><em><strong><span class="mjpage">[2 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = 9 - {x^2}">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>9</mn>
<mo>−<!-- − --></mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \in \mathbb{R}">
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>The following diagram shows part of the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span>.</p>
<p style="text-align: center;"><img 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"></p>
<p>Rectangle PQRS is drawn with P and Q on the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis and R and S on the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span>.</p>
<p>Let OP = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span>.</p>
</div>
<div class="specification">
<p>Consider another function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g(x) = {\left( {x - 3} \right)^2} + k">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>3</mn>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>k</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \in \mathbb{R}">
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-intercepts of the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the area of PQRS is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="18b - 2{b^3}"> <mn>18</mn> <mi>b</mi> <mo>−</mo> <mn>2</mn> <mrow> <msup> <mi>b</mi> <mn>3</mn> </msup> </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>Hence find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b"> <mi>b</mi> </math></span> such that the area of PQRS is a maximum.</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>Show that when the graphs of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g"> <mi>g</mi> </math></span> intersect, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2{x^2} - 6x + k = 0"> <mn>2</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>6</mn> <mi>x</mi> <mo>+</mo> <mi>k</mi> <mo>=</mo> <mn>0</mn> </math></span>.</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>Given that the graphs of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g"> <mi>g</mi> </math></span> intersect only once, find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k"> <mi>k</mi> </math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">e.</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>valid approach <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <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>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="9 - {x^2} = 0"> <mn>9</mn> <mo>−</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mn>0</mn> </math></span> , one correct solution</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = - 3"> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mn>3</mn> </math></span>, 3 (accept (3, 0), (−3, 0)) <em><strong>A1 N2</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>valid approach <em><strong>(M1)</strong></em></p>
<p><em>eg</em> height = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(b)"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </math></span>, base = 2(OP) or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2b"> <mn>2</mn> <mi>b</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2b\left( {9 - {x^2}} \right)"> <mn>2</mn> <mi>b</mi> <mrow> <mo>(</mo> <mrow> <mn>9</mn> <mo>−</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2b \times f(b)"> <mn>2</mn> <mi>b</mi> <mo>×</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </math></span></p>
<p>correct working that clearly leads to given answer <em><strong>A1</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2b\left( {9 - {b^2}} \right)"> <mn>2</mn> <mi>b</mi> <mrow> <mo>(</mo> <mrow> <mn>9</mn> <mo>−</mo> <mrow> <msup> <mi>b</mi> <mn>2</mn> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p>Note: Do not accept sloppy notation <em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2b \times 9 - {b^2}"> <mn>2</mn> <mi>b</mi> <mo>×</mo> <mn>9</mn> <mo>−</mo> <mrow> <msup> <mi>b</mi> <mn>2</mn> </msup> </mrow> </math></span>.</p>
<p>area = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="18b - 2{b^3}"> <mn>18</mn> <mi>b</mi> <mo>−</mo> <mn>2</mn> <mrow> <msup> <mi>b</mi> <mn>3</mn> </msup> </mrow> </math></span> <em><strong>AG N0</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>setting derivative = 0 (seen anywhere) <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A' = 0"> <msup> <mi>A</mi> <mo>′</mo> </msup> <mo>=</mo> <mn>0</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left[ {18b - 2{b^3}} \right]^\prime } = 0"> <mrow> <msup> <mrow> <mo>[</mo> <mrow> <mn>18</mn> <mi>b</mi> <mo>−</mo> <mn>2</mn> <mrow> <msup> <mi>b</mi> <mn>3</mn> </msup> </mrow> </mrow> <mo>]</mo> </mrow> <mi mathvariant="normal">′</mi> </msup> </mrow> <mo>=</mo> <mn>0</mn> </math></span> </p>
<p>correct derivative (must be in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b"> <mi>b</mi> </math></span> only) (seen anywhere) <em><strong>A2</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="18 - 6{b^2}"> <mn>18</mn> <mo>−</mo> <mn>6</mn> <mrow> <msup> <mi>b</mi> <mn>2</mn> </msup> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2b\left( { - 2b} \right) + \left( {9 - {b^2}} \right) \times 2"> <mn>2</mn> <mi>b</mi> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>2</mn> <mi>b</mi> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mn>9</mn> <mo>−</mo> <mrow> <msup> <mi>b</mi> <mn>2</mn> </msup> </mrow> </mrow> <mo>)</mo> </mrow> <mo>×</mo> <mn>2</mn> </math></span></p>
<p>correct working <em><strong>(A1)</strong></em></p>
<p><em>eg </em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="6{b^2} = 18"> <mn>6</mn> <mrow> <msup> <mi>b</mi> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mn>18</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b = \pm \sqrt 3 "> <mi>b</mi> <mo>=</mo> <mo>±</mo> <msqrt> <mn>3</mn> </msqrt> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b = \sqrt 3 "> <mi>b</mi> <mo>=</mo> <msqrt> <mn>3</mn> </msqrt> </math></span> <em><strong>A1 N3</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid approach <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f = g"> <mi>f</mi> <mo>=</mo> <mi>g</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="9 - {x^2} = {\left( {x - 3} \right)^2} + k"> <mn>9</mn> <mo>−</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mn>3</mn> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mi>k</mi> </math></span> </p>
<p>correct working <em><strong>(A1)</strong></em></p>
<p><em>eg </em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="9 - {x^2} = {x^2} - 6x + 9 + k"> <mn>9</mn> <mo>−</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>6</mn> <mi>x</mi> <mo>+</mo> <mn>9</mn> <mo>+</mo> <mi>k</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="9 - {x^2} - {x^2} + 6x - 9 - k = 0"> <mn>9</mn> <mo>−</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>6</mn> <mi>x</mi> <mo>−</mo> <mn>9</mn> <mo>−</mo> <mi>k</mi> <mo>=</mo> <mn>0</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2{x^2} - 6x + k = 0"> <mn>2</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>6</mn> <mi>x</mi> <mo>+</mo> <mi>k</mi> <mo>=</mo> <mn>0</mn> </math></span> <em><strong>AG N0</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1 (discriminant)</strong></p>
<p>recognizing to use discriminant (seen anywhere) <em><strong> (M1)</strong></em></p>
<p><em>eg</em> Δ, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{b^2} - 4ac"> <mrow> <msup> <mi>b</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>4</mn> <mi>a</mi> <mi>c</mi> </math></span></p>
<p>discriminant = 0 (seen anywhere) <em><strong>M1</strong></em></p>
<p>correct substitution into discriminant (do not accept only in quadratic formula) <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( { - 6} \right)^2} - 4\left( 2 \right)\left( k \right)"> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>6</mn> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>4</mn> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( 6 \right)^2} - 4\left( 2 \right)\left( k \right)"> <mrow> <msup> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>4</mn> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </math></span></p>
<p>correct working <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="36 - 8k = 0"> <mn>36</mn> <mo>−</mo> <mn>8</mn> <mi>k</mi> <mo>=</mo> <mn>0</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="8k = 36"> <mn>8</mn> <mi>k</mi> <mo>=</mo> <mn>36</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = \frac{{36}}{8}\,\,\,\,\left( { = \frac{9}{2}{\text{,}}\,\,4.5} \right)"> <mi>k</mi> <mo>=</mo> <mfrac> <mrow> <mn>36</mn> </mrow> <mn>8</mn> </mfrac> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mfrac> <mn>9</mn> <mn>2</mn> </mfrac> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mn>4.5</mn> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1 N2</strong></em></p>
<p> </p>
<p><strong>METHOD 2 (completing the square)</strong></p>
<p>valid approach to complete the square <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\left( {{x^2} - 3x + \frac{9}{4}} \right) = - k + \frac{{18}}{4}"> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mfrac> <mn>9</mn> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mi>k</mi> <mo>+</mo> <mfrac> <mrow> <mn>18</mn> </mrow> <mn>4</mn> </mfrac> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} - 3x + \frac{9}{4} - \frac{9}{4} + \frac{k}{2} = 0"> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mfrac> <mn>9</mn> <mn>4</mn> </mfrac> <mo>−</mo> <mfrac> <mn>9</mn> <mn>4</mn> </mfrac> <mo>+</mo> <mfrac> <mi>k</mi> <mn>2</mn> </mfrac> <mo>=</mo> <mn>0</mn> </math></span></p>
<p>correct working <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2{\left( {x - \frac{3}{2}} \right)^2} = - k + \frac{{18}}{4}"> <mn>2</mn> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mo>−</mo> <mi>k</mi> <mo>+</mo> <mfrac> <mrow> <mn>18</mn> </mrow> <mn>4</mn> </mfrac> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {x - \frac{3}{2}} \right)^2} - \frac{9}{4} + \frac{k}{2} = 0"> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mfrac> <mn>9</mn> <mn>4</mn> </mfrac> <mo>+</mo> <mfrac> <mi>k</mi> <mn>2</mn> </mfrac> <mo>=</mo> <mn>0</mn> </math></span></p>
<p>recognizing condition for one solution <em><strong>M1</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {x - \frac{3}{2}} \right)^2} = 0"> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mn>0</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \frac{9}{4} + \frac{k}{2} = 0"> <mo>−</mo> <mfrac> <mn>9</mn> <mn>4</mn> </mfrac> <mo>+</mo> <mfrac> <mi>k</mi> <mn>2</mn> </mfrac> <mo>=</mo> <mn>0</mn> </math></span></p>
<p>correct working <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - k = \frac{{18}}{4}"> <mo>−</mo> <mi>k</mi> <mo>=</mo> <mfrac> <mrow> <mn>18</mn> </mrow> <mn>4</mn> </mfrac> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{k}{2} = \frac{9}{4}"> <mfrac> <mi>k</mi> <mn>2</mn> </mfrac> <mo>=</mo> <mfrac> <mn>9</mn> <mn>4</mn> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = \frac{{18}}{4}\,\,\,\,\left( { = \frac{9}{2}{\text{,}}\,\,4.5} \right)"> <mi>k</mi> <mo>=</mo> <mfrac> <mrow> <mn>18</mn> </mrow> <mn>4</mn> </mfrac> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mfrac> <mn>9</mn> <mn>2</mn> </mfrac> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mn>4.5</mn> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1 N2</strong></em></p>
<p> </p>
<p><strong>METHOD 3 (using vertex)<br></strong></p>
<p>valid approach to find vertex (seen anywhere) <em><strong>M1</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {2{x^2} - 6x + k} \right)^\prime } = 0"> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>6</mn> <mi>x</mi> <mo>+</mo> <mi>k</mi> </mrow> <mo>)</mo> </mrow> <mi mathvariant="normal">′</mi> </msup> </mrow> <mo>=</mo> <mn>0</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \frac{b}{{2a}}"> <mo>−</mo> <mfrac> <mi>b</mi> <mrow> <mn>2</mn> <mi>a</mi> </mrow> </mfrac> </math></span></p>
<p>correct working <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {2{x^2} - 6x + k} \right)^\prime } = 4x - 6"> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>6</mn> <mi>x</mi> <mo>+</mo> <mi>k</mi> </mrow> <mo>)</mo> </mrow> <mi mathvariant="normal">′</mi> </msup> </mrow> <mo>=</mo> <mn>4</mn> <mi>x</mi> <mo>−</mo> <mn>6</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \frac{{\left( { - 6} \right)}}{{2\left( 2 \right)}}"> <mo>−</mo> <mfrac> <mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>6</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{6}{4}\,\,\left( { = \frac{3}{2}} \right)"> <mi>x</mi> <mo>=</mo> <mfrac> <mn>6</mn> <mn>4</mn> </mfrac> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>(A1)</strong></em></p>
<p>correct substitution <em><strong>(A1)</strong></em></p>
<p><em>eg </em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2{\left( {\frac{3}{2}} \right)^2} - 6\left( {\frac{3}{2}} \right) + k = 0"> <mn>2</mn> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>6</mn> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>k</mi> <mo>=</mo> <mn>0</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = \frac{{18}}{4}\,\,\,\,\left( { = \frac{9}{2}{\text{,}}\,\,4.5} \right)"> <mi>k</mi> <mo>=</mo> <mfrac> <mrow> <mn>18</mn> </mrow> <mn>4</mn> </mfrac> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mfrac> <mn>9</mn> <mn>2</mn> </mfrac> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mn>4.5</mn> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1 N2</strong></em></p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {x{{\text{e}}^{{x^2} - 1}}{\text{d}}x} ">
<mo>∫</mo>
<mrow>
<mi>x</mi>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</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>Find <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>, given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f’(x) = x{{\text{e}}^{{x^2} - 1}}">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>x</mi>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f( - 1) = 3">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>3</mn>
</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>valid approach to set up integration by substitution/inspection <strong><em>(M1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u = {x^2} - 1,{\text{ d}}u = 2x,{\text{ }}\int {2x{{\text{e}}^{{x^2} - 1}}{\text{d}}x} ">
<mi>u</mi>
<mo>=</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
<mo>,</mo>
<mrow>
<mtext> d</mtext>
</mrow>
<mi>u</mi>
<mo>=</mo>
<mn>2</mn>
<mi>x</mi>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo>∫</mo>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</math></span></p>
<p>correct expression <strong><em>(A1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2}\int {2x{{\text{e}}^{{x^2} - 1}}{\text{d}}x,{\text{ }}\frac{1}{2}\int {{{\text{e}}^u}{\text{d}}u} } ">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>∫</mo>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>∫</mo>
<mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mi>u</mi>
</msup>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>u</mi>
</mrow>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2}{{\text{e}}^{{x^2} - 1}} + c">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mi>c</mi>
</math></span> <strong><em>A2</em></strong> <strong><em>N4</em></strong></p>
<p> </p>
<p><strong>Notes: </strong>Award <strong><em>A1</em> </strong>if missing “<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" + c">
<mo>+</mo>
<mi>c</mi>
</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>substituting <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> into <strong>their </strong>answer from (a) <strong><em>(M1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2}{{\text{e}}^0},{\text{ }}\frac{1}{2}{{\text{e}}^{1 - 1}} = 3">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mn>0</mn>
</msup>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mn>3</mn>
</math></span></p>
<p>correct working <strong><em>(A1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2} + c = 3,{\text{ }}c = 2.5">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>+</mo>
<mi>c</mi>
<mo>=</mo>
<mn>3</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>c</mi>
<mo>=</mo>
<mn>2.5</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = \frac{1}{2}{{\text{e}}^{{x^2} - 1}} + 2.5">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mn>2.5</mn>
</math></span> <strong><em>A1</em></strong> <strong><em>N2</em></strong></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>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {\left( {{x^3} + x} \right)^{\frac{3}{2}}}">
<mi>y</mi>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mfrac>
<mn>3</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>Consider the functions <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = \sqrt {{x^3} + x} ">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>x</mi>
</msqrt>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( x \right) = 6 - 3{x^2}\sqrt {{x^3} + x} ">
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>6</mn>
<mo>−<!-- − --></mo>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>x</mi>
</msqrt>
</math></span>, for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> ≥ 0.</p>
<p>The graphs of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g">
<mi>g</mi>
</math></span> are shown in the following diagram.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p>The shaded region <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="R">
<mi>R</mi>
</math></span> is enclosed by the graphs of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g">
<mi>g</mi>
</math></span>, the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-axis and <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>.</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>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {\left( {3{x^2} + 1} \right)\sqrt {{x^3} + x} } \,{\text{d}}x"> <mo>∫</mo> <mrow> <mrow> <mo>(</mo> <mrow> <mn>3</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <msqrt> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> <mo>+</mo> <mi>x</mi> </msqrt> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down an expression for the area of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="R"> <mi>R</mi> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the exact area of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="R"> <mi>R</mi> </math></span>.</p>
<div class="marks">[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 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>evidence of choosing chain rule <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{{\text{d}}y}}{{{\text{d}}u}} \times \frac{{{\text{d}}u}}{{{\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> <mo>=</mo> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>u</mi> </mrow> </mfrac> <mo>×</mo> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>u</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u = {x^3} + x"> <mi>u</mi> <mo>=</mo> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> <mo>+</mo> <mi>x</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u' = 3{x^2} + 1"> <msup> <mi>u</mi> <mo>′</mo> </msup> <mo>=</mo> <mn>3</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>1</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{3}{2}{\left( {{x^3} + x} \right)^{\frac{1}{2}}}\left( {3{x^2} + 1} \right)\,\,\,\left( { = \frac{3}{2}\sqrt {{x^3} + x} \left( {3{x^2} + 1} \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> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> <mo>+</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <mrow> <mn>3</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> <msqrt> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> <mo>+</mo> <mi>x</mi> </msqrt> <mrow> <mo>(</mo> <mrow> <mn>3</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A2 N3</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>integrating by inspection from (a) or by substitution <em><strong>(M1)</strong></em></p>
<p>eg <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{2}{3}\int {\frac{3}{2}} \left( {3{x^2} + 1} \right)\sqrt {{x^3} + x} \,{\text{d}}x"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> <mo>∫</mo> <mrow> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>3</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <msqrt> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> <mo>+</mo> <mi>x</mi> </msqrt> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u = {x^3} + x"> <mi>u</mi> <mo>=</mo> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> <mo>+</mo> <mi>x</mi> </math></span>, <span style="background-color: #ffffff;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}u}}{{{\text{d}}x}} = 3{x^2} + 1"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>u</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mn>3</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>1</mn> </math></span></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {{u^{\frac{1}{2}}}} "> <mo>∫</mo> <mrow> <mrow> <msup> <mi>u</mi> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{u^{\frac{3}{2}}}}}{{1.5}}"> <mfrac> <mrow> <mrow> <msup> <mi>u</mi> <mrow> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mrow> <mn>1.5</mn> </mrow> </mfrac> </math></span></p>
<p>correct integrated expression in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> <em><strong>A2 N3</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{2}{3}{\left( {{x^3} + x} \right)^{\frac{3}{2}}} + C"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> <mo>+</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> <mo>+</mo> <mi>C</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{{\left( {{x^3} + x} \right)}^{1.5}}}}{{1.5}} + C"> <mfrac> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> <mo>+</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>1.5</mn> </mrow> </msup> </mrow> </mrow> <mrow> <mn>1.5</mn> </mrow> </mfrac> <mo>+</mo> <mi>C</mi> </math></span></p>
<p><em><strong>[3 marks]</strong></em></p>
<p> </p>
<p> </p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>integrating and subtracting functions (in any order) <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {g - f} "> <mo>∫</mo> <mrow> <mi>g</mi> <mo>−</mo> <mi>f</mi> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {f - \int g } "> <mo>∫</mo> <mrow> <mi>f</mi> <mo>−</mo> <mo>∫</mo> <mi>g</mi> </mrow> </math></span></p>
<p>correct integral (including limits, accept absence of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{d}}x}"> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </math></span>) <em><strong>A1 N2</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^1 {\left( {g - f} \right)} \,{\text{d}}x"> <msubsup> <mo>∫</mo> <mn>0</mn> <mn>1</mn> </msubsup> <mrow> <mrow> <mo>(</mo> <mrow> <mi>g</mi> <mo>−</mo> <mi>f</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^1 {6 - 3{x^2}\sqrt {{x^3} + x} - \sqrt {{x^3} + x} } \,{\text{d}}x"> <msubsup> <mo>∫</mo> <mn>0</mn> <mn>1</mn> </msubsup> <mrow> <mn>6</mn> <mo>−</mo> <mn>3</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <msqrt> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> <mo>+</mo> <mi>x</mi> </msqrt> <mo>−</mo> <msqrt> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> <mo>+</mo> <mi>x</mi> </msqrt> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^1 {g\left( x \right) - } \int_0^1 {f\left( x \right)} "> <msubsup> <mo>∫</mo> <mn>0</mn> <mn>1</mn> </msubsup> <mrow> <mi>g</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>−</mo> </mrow> <msubsup> <mo>∫</mo> <mn>0</mn> <mn>1</mn> </msubsup> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> </math></span></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 <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt {{x^3} + x} "> <msqrt> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> <mo>+</mo> <mi>x</mi> </msqrt> </math></span> is a common factor (seen anywhere, may be seen in part (c)) <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( { - 3{x^2} - 1} \right)\sqrt {{x^3} + x} "> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>3</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <msqrt> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> <mo>+</mo> <mi>x</mi> </msqrt> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {6 - \left( {3{x^2} + 1} \right)} \sqrt {{x^3} + x} "> <mo>∫</mo> <mrow> <mn>6</mn> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mn>3</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <msqrt> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> <mo>+</mo> <mi>x</mi> </msqrt> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {3{x^2} - 1} \right)\sqrt {{x^3} + x} "> <mrow> <mo>(</mo> <mrow> <mn>3</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <msqrt> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> <mo>+</mo> <mi>x</mi> </msqrt> </math></span></p>
<p>correct integration <em><strong>(A1)(A1)</strong></em></p>
<p>eg <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="6x - \frac{2}{3}{\left( {{x^3} + x} \right)^{\frac{3}{2}}}"> <mn>6</mn> <mi>x</mi> <mo>−</mo> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> <mo>+</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> </math></span></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong> </em>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="6x"> <mn>6</mn> <mi>x</mi> </math></span> and award <em><strong>A1</strong> </em>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \frac{2}{3}{\left( {{x^3} + x} \right)^{\frac{3}{2}}}"> <mo>−</mo> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> <mo>+</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> </math></span>.</p>
<p>substituting limits into <strong>their</strong> integrated function and subtracting (in any order) <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="6 - \frac{2}{3}{\left( {{1^3} + 1} \right)^{\frac{3}{2}}}"> <mn>6</mn> <mo>−</mo> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mn>1</mn> <mn>3</mn> </msup> </mrow> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 - \left[ {6 - \frac{2}{3}{{\left( {{1^3} + 1} \right)}^{\frac{3}{2}}}} \right]"> <mn>0</mn> <mo>−</mo> <mrow> <mo>[</mo> <mrow> <mn>6</mn> <mo>−</mo> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mn>1</mn> <mn>3</mn> </msup> </mrow> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mo>]</mo> </mrow> </math></span></p>
<p>correct working <em><strong> (A1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="6 - \frac{2}{3} \times 2\sqrt 2 "> <mn>6</mn> <mo>−</mo> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> <mo>×</mo> <mn>2</mn> <msqrt> <mn>2</mn> </msqrt> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="6 - \frac{2}{3} \times \sqrt 4 \times \sqrt 2 "> <mn>6</mn> <mo>−</mo> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> <mo>×</mo> <msqrt> <mn>4</mn> </msqrt> <mo>×</mo> <msqrt> <mn>2</mn> </msqrt> </math></span></p>
<p>area of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="R = 6 - \frac{{4\sqrt 2 }}{3}\,\,\,\left( { = 6 - \frac{2}{3}\sqrt 8 {\text{,}}\,\,\,6 - \frac{2}{3} \times {2^{\frac{3}{2}}}{\text{,}}\,\,\,\frac{{18 - 4\sqrt 2 }}{3}} \right)"> <mi>R</mi> <mo>=</mo> <mn>6</mn> <mo>−</mo> <mfrac> <mrow> <mn>4</mn> <msqrt> <mn>2</mn> </msqrt> </mrow> <mn>3</mn> </mfrac> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mn>6</mn> <mo>−</mo> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> <msqrt> <mn>8</mn> </msqrt> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mn>6</mn> <mo>−</mo> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> <mo>×</mo> <mrow> <msup> <mn>2</mn> <mrow> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mfrac> <mrow> <mn>18</mn> <mo>−</mo> <mn>4</mn> <msqrt> <mn>2</mn> </msqrt> </mrow> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1 N3</strong></em></p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = 6{x^2} - 3x">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>6</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>3</mn>
<mi>x</mi>
</math></span>. The graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> 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="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {\left( {6{x^2} - 3x} \right){\text{d}}x} ">
<mo>∫</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>6</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area of the region enclosed by the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span>, the <em>x</em>-axis and the lines <em>x</em> = 1 and <em>x</em> = 2 .</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2{x^3} - \frac{{3{x^2}}}{2} + c\,\,\,\left( {{\text{accept}}\,\,\frac{{6{x^3}}}{3} - \frac{{3{x^2}}}{2} + c} \right)">
<mn>2</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mfrac>
<mrow>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mn>2</mn>
</mfrac>
<mo>+</mo>
<mi>c</mi>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>accept</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mfrac>
<mrow>
<mn>6</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mn>3</mn>
</mfrac>
<mo>−</mo>
<mfrac>
<mrow>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mn>2</mn>
</mfrac>
<mo>+</mo>
<mi>c</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1A1 N2</strong></em></p>
<p><strong>Notes:</strong> Award <em><strong>A1A0</strong></em> for both correct terms if +<em>c</em> is omitted.<br>Award <em><strong>A1A0</strong></em> for one correct term <em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2{x^3} + c">
<mn>2</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>c</mi>
</math></span>.<br>Award <em><strong>A1A0</strong></em> if both terms are correct, but candidate attempts further working to solve for <em>c</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>substitution of limits or function <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_1^2 {f\left( x \right)} \,{\text{d}}x,\,\,\left[ {2{x^3} - \frac{{3{x^2}}}{2}} \right]_1^2">
<msubsup>
<mo>∫</mo>
<mn>1</mn>
<mn>2</mn>
</msubsup>
<mrow>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<msubsup>
<mrow>
<mo>[</mo>
<mrow>
<mn>2</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mfrac>
<mrow>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
<mo>]</mo>
</mrow>
<mn>1</mn>
<mn>2</mn>
</msubsup>
</math></span></p>
<p>substituting limits into their integrated function and subtracting <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{6 \times {2^3}}}{3} - \frac{{3 \times {2^2}}}{2} - \left( {\frac{{6 \times {1^3}}}{3} + \frac{{3 \times {1^2}}}{2}} \right)">
<mfrac>
<mrow>
<mn>6</mn>
<mo>×</mo>
<mrow>
<msup>
<mn>2</mn>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mn>3</mn>
</mfrac>
<mo>−</mo>
<mfrac>
<mrow>
<mn>3</mn>
<mo>×</mo>
<mrow>
<msup>
<mn>2</mn>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mn>2</mn>
</mfrac>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>6</mn>
<mo>×</mo>
<mrow>
<msup>
<mn>1</mn>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mn>3</mn>
</mfrac>
<mo>+</mo>
<mfrac>
<mrow>
<mn>3</mn>
<mo>×</mo>
<mrow>
<msup>
<mn>1</mn>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><strong>Note:</strong> Award <em><strong>M0</strong> </em>if substituted into original function.</p>
<p>correct working <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{6 \times 8}}{3} - \frac{{3 \times 4}}{2} - \frac{{6 \times 1}}{3} + \frac{{3 \times 1}}{2},\,\,\left( {16 - 6} \right) - \left( {2 - \frac{3}{2}} \right)">
<mfrac>
<mrow>
<mn>6</mn>
<mo>×</mo>
<mn>8</mn>
</mrow>
<mn>3</mn>
</mfrac>
<mo>−</mo>
<mfrac>
<mrow>
<mn>3</mn>
<mo>×</mo>
<mn>4</mn>
</mrow>
<mn>2</mn>
</mfrac>
<mo>−</mo>
<mfrac>
<mrow>
<mn>6</mn>
<mo>×</mo>
<mn>1</mn>
</mrow>
<mn>3</mn>
</mfrac>
<mo>+</mo>
<mfrac>
<mrow>
<mn>3</mn>
<mo>×</mo>
<mn>1</mn>
</mrow>
<mn>2</mn>
</mfrac>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mn>16</mn>
<mo>−</mo>
<mn>6</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mo>−</mo>
<mfrac>
<mn>3</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{19}}{2}">
<mfrac>
<mrow>
<mn>19</mn>
</mrow>
<mn>2</mn>
</mfrac>
</math></span> <em><strong>A1 N3</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The values of the functions <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g">
<mi>g</mi>
</math></span> and their derivatives for <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 = 8">
<mi>x</mi>
<mo>=</mo>
<mn>8</mn>
</math></span> are shown in the following table.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-11_om_16.42.43.png" alt="M17/5/MATME/SP1/ENG/TZ2/06"></p>
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h(x) = f(x)g(x)">
<mi>h</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</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="h(1)">
<mi>h</mi>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h'(8)">
<msup>
<mi>h</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mn>8</mn>
<mo stretchy="false">)</mo>
</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>expressing <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h(1)">
<mi>h</mi>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</math></span> as a product of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(1)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g(1)">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(A1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(1) \times g(1),{\text{ }}2(9)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>×</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>2</mn>
<mo stretchy="false">(</mo>
<mn>9</mn>
<mo stretchy="false">)</mo>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h(1) = 18">
<mi>h</mi>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>18</mn>
</math></span> <strong><em>A1</em></strong> <strong><em>N2</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>attempt to use product rule (do <strong>not </strong>accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h’ = f' \times g’">
<msup>
<mi>h</mi>
<mo>′</mo>
</msup>
<mo>=</mo>
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mo>×</mo>
<msup>
<mi>g</mi>
<mo>′</mo>
</msup>
</math></span>) <strong><em>(M1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h’ = fg' + gf',{\text{ }}h'(8) = f'(8)g(8) + g’(8)f(8)">
<msup>
<mi>h</mi>
<mo>′</mo>
</msup>
<mo>=</mo>
<mi>f</mi>
<msup>
<mi>g</mi>
<mo>′</mo>
</msup>
<mo>+</mo>
<mi>g</mi>
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<msup>
<mi>h</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mn>8</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mn>8</mn>
<mo stretchy="false">)</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mn>8</mn>
<mo stretchy="false">)</mo>
<mo>+</mo>
<msup>
<mi>g</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mn>8</mn>
<mo stretchy="false">)</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mn>8</mn>
<mo stretchy="false">)</mo>
</math></span></p>
<p>correct substitution of values into product rule <strong><em>(A1) </em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h’(8) = 4(5) + 2( - 3),{\text{ }} - 6 + 20">
<msup>
<mi>h</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mn>8</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>4</mn>
<mo stretchy="false">(</mo>
<mn>5</mn>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mn>2</mn>
<mo stretchy="false">(</mo>
<mo>−</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo>−</mo>
<mn>6</mn>
<mo>+</mo>
<mn>20</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h’(8) = 14">
<msup>
<mi>h</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mn>8</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>14</mn>
</math></span> <strong><em>A1 N2</em></strong></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>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = \cos x">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>cos</mi>
<mo><!-- --></mo>
<mi>x</mi>
</math></span>.</p>
</div>
<div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g(x) = {x^k}">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<msup>
<mi>x</mi>
<mi>k</mi>
</msup>
</mrow>
</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>
<msup>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = 21">
<mi>k</mi>
<mo>=</mo>
<mn>21</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h(x) = \left( {{f^{(19)}}(x) \times {g^{(19)}}(x)} \right)">
<mi>h</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mo stretchy="false">(</mo>
<mn>19</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>×<!-- × --></mo>
<mrow>
<msup>
<mi>g</mi>
<mrow>
<mo stretchy="false">(</mo>
<mn>19</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>(i) Find the first four derivatives of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span>.</p>
<p>(ii) Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f^{(19)}}(x)">
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mo stretchy="false">(</mo>
<mn>19</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</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>(i) Find the first three derivatives of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g(x)">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span>.</p>
<p>(ii) Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{g^{(19)}}(x) = \frac{{k!}}{{(k - p)!}}({x^{k - 19}})">
<mrow>
<msup>
<mi>g</mi>
<mrow>
<mo stretchy="false">(</mo>
<mn>19</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mrow>
<mi>k</mi>
<mo>!</mo>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>−</mo>
<mi>p</mi>
<mo stretchy="false">)</mo>
<mo>!</mo>
</mrow>
</mfrac>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mi>k</mi>
<mo>−</mo>
<mn>19</mn>
</mrow>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</math></span>, find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span>.</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>(i) Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h'(x)">
<msup>
<mi>h</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span>.</p>
<p>(ii) Hence, show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h'(\pi ) = \frac{{ - 21!}}{2}{\pi ^2}">
<msup>
<mi>h</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>π</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mrow>
<mo>−</mo>
<mn>21</mn>
<mo>!</mo>
</mrow>
<mn>2</mn>
</mfrac>
<mrow>
<msup>
<mi>π</mi>
<mn>2</mn>
</msup>
</mrow>
</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>(i) <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'(x) = - \sin x,{\text{ }}f''(x) = - \cos x,{\text{ }}{f^{(3)}}(x) = \sin x,{\text{ }}{f^{(4)}}(x) = \cos x">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>−</mo>
<mi>sin</mi>
<mo></mo>
<mi>x</mi>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<msup>
<mi>f</mi>
<mo>″</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>−</mo>
<mi>cos</mi>
<mo></mo>
<mi>x</mi>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mo stretchy="false">(</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>sin</mi>
<mo></mo>
<mi>x</mi>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mo stretchy="false">(</mo>
<mn>4</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>cos</mi>
<mo></mo>
<mi>x</mi>
</math></span> <strong><em>A2 N2</em></strong></p>
<p>(ii) valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span>recognizing that 19 is one less than a multiple of 4, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f^{(19)}}(x) = {f^{(3)}}(x)">
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mo stretchy="false">(</mo>
<mn>19</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mo stretchy="false">(</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f^{(19)}}(x) = \sin x">
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mo stretchy="false">(</mo>
<mn>19</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>sin</mi>
<mo></mo>
<mi>x</mi>
</math></span> <strong><em>A1 N2</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(i) <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'(x) = k{x^{k - 1}}">
<msup>
<mi>g</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>k</mi>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mi>k</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g''(x) = k(k - 1){x^{k - 2}},{\text{ }}{g^{(3)}}(x) = k(k - 1)(k - 2){x^{k - 3}}">
<msup>
<mi>g</mi>
<mo>″</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>k</mi>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mi>k</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<msup>
<mi>g</mi>
<mrow>
<mo stretchy="false">(</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>k</mi>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>−</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mi>k</mi>
<mo>−</mo>
<mn>3</mn>
</mrow>
</msup>
</mrow>
</math></span> <strong><em>A1A1 N2</em></strong></p>
<p>(ii) <strong>METHOD 1</strong></p>
<p>correct working that leads to the correct answer, involving the correct expression for the 19th derivative <strong><em>A2</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k(k - 1)(k - 2) \ldots (k - 18) \times \frac{{(k - 19)!}}{{(k - 19)!}},{{\text{ }}_k}{P_{19}}">
<mi>k</mi>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>−</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
<mo>…</mo>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>−</mo>
<mn>18</mn>
<mo stretchy="false">)</mo>
<mo>×</mo>
<mfrac>
<mrow>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>−</mo>
<mn>19</mn>
<mo stretchy="false">)</mo>
<mo>!</mo>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>−</mo>
<mn>19</mn>
<mo stretchy="false">)</mo>
<mo>!</mo>
</mrow>
</mfrac>
<mo>,</mo>
<mrow>
<msub>
<mrow>
<mtext> </mtext>
</mrow>
<mi>k</mi>
</msub>
</mrow>
<mrow>
<msub>
<mi>P</mi>
<mrow>
<mn>19</mn>
</mrow>
</msub>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p = 19">
<mi>p</mi>
<mo>=</mo>
<mn>19</mn>
</math></span> (accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{k!}}{{(k - 19)!}}{x^{k - 19}}">
<mfrac>
<mrow>
<mi>k</mi>
<mo>!</mo>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>−</mo>
<mn>19</mn>
<mo stretchy="false">)</mo>
<mo>!</mo>
</mrow>
</mfrac>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mi>k</mi>
<mo>−</mo>
<mn>19</mn>
</mrow>
</msup>
</mrow>
</math></span>) <strong><em>A1 N1</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p>correct working involving recognizing patterns in coefficients of first three derivatives (may be seen in part (b)(i)) leading to a general rule for 19th coefficient <strong><em>A2</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'' = 2!\left( {\begin{array}{*{20}{c}} k \\ 2 \end{array}} \right),{\text{ }}k(k - 1)(k - 2) = \frac{{k!}}{{(k - 3)!}},{\text{ }}{g^{(3)}}(x){ = _k}{P_3}({x^{k - 3}})">
<msup>
<mi>g</mi>
<mo>″</mo>
</msup>
<mo>=</mo>
<mn>2</mn>
<mo>!</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>k</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>k</mi>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>−</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mrow>
<mi>k</mi>
<mo>!</mo>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>−</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
<mo>!</mo>
</mrow>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<msup>
<mi>g</mi>
<mrow>
<mo stretchy="false">(</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mrow>
<msub>
<mo>=</mo>
<mi>k</mi>
</msub>
</mrow>
<mrow>
<msub>
<mi>P</mi>
<mn>3</mn>
</msub>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mi>k</mi>
<mo>−</mo>
<mn>3</mn>
</mrow>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{g^{(19)}}(x) = 19!\left( {\begin{array}{*{20}{c}} k \\ {19} \end{array}} \right),{\text{ }}19! \times \frac{{k!}}{{(k - 19)! \times 19!}},{{\text{ }}_k}{P_{19}}">
<mrow>
<msup>
<mi>g</mi>
<mrow>
<mo stretchy="false">(</mo>
<mn>19</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>19</mn>
<mo>!</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>k</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>19</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>19</mn>
<mo>!</mo>
<mo>×</mo>
<mfrac>
<mrow>
<mi>k</mi>
<mo>!</mo>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>−</mo>
<mn>19</mn>
<mo stretchy="false">)</mo>
<mo>!</mo>
<mo>×</mo>
<mn>19</mn>
<mo>!</mo>
</mrow>
</mfrac>
<mo>,</mo>
<mrow>
<msub>
<mrow>
<mtext> </mtext>
</mrow>
<mi>k</mi>
</msub>
</mrow>
<mrow>
<msub>
<mi>P</mi>
<mrow>
<mn>19</mn>
</mrow>
</msub>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p = 19">
<mi>p</mi>
<mo>=</mo>
<mn>19</mn>
</math></span> (accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{k!}}{{(k - 19)!}}{x^{k - 19}}">
<mfrac>
<mrow>
<mi>k</mi>
<mo>!</mo>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>−</mo>
<mn>19</mn>
<mo stretchy="false">)</mo>
<mo>!</mo>
</mrow>
</mfrac>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mi>k</mi>
<mo>−</mo>
<mn>19</mn>
</mrow>
</msup>
</mrow>
</math></span>) <strong><em>A1 N1</em></strong></p>
<p><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(i) valid approach using product rule <strong><em>(M1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="uv' + vu',{\text{ }}{f^{(19)}}{g^{(20)}} + {f^{(20)}}{g^{(19)}}">
<mi>u</mi>
<msup>
<mi>v</mi>
<mo>′</mo>
</msup>
<mo>+</mo>
<mi>v</mi>
<msup>
<mi>u</mi>
<mo>′</mo>
</msup>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mo stretchy="false">(</mo>
<mn>19</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
<mrow>
<msup>
<mi>g</mi>
<mrow>
<mo stretchy="false">(</mo>
<mn>20</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mo stretchy="false">(</mo>
<mn>20</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
<mrow>
<msup>
<mi>g</mi>
<mrow>
<mo stretchy="false">(</mo>
<mn>19</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
</math></span></p>
<p>correct 20th derivatives (must be seen in product rule) <strong><em>(A1)(A1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{g^{(20)}}(x) = \frac{{21!}}{{(21 - 20)!}}x,{\text{ }}{f^{(20)}}(x) = \cos x">
<mrow>
<msup>
<mi>g</mi>
<mrow>
<mo stretchy="false">(</mo>
<mn>20</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mrow>
<mn>21</mn>
<mo>!</mo>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mn>21</mn>
<mo>−</mo>
<mn>20</mn>
<mo stretchy="false">)</mo>
<mo>!</mo>
</mrow>
</mfrac>
<mi>x</mi>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mo stretchy="false">(</mo>
<mn>20</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>cos</mi>
<mo></mo>
<mi>x</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h'(x) = \sin x(21!x) + \cos x\left( {\frac{{21!}}{2}{x^2}} \right){\text{ }}\left( {{\text{accept }}\sin x\left( {\frac{{21!}}{{1!}}x} \right) + \cos x\left( {\frac{{21!}}{{2!}}{x^2}} \right)} \right)">
<msup>
<mi>h</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>sin</mi>
<mo></mo>
<mi>x</mi>
<mo stretchy="false">(</mo>
<mn>21</mn>
<mo>!</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mi>cos</mi>
<mo></mo>
<mi>x</mi>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>21</mn>
<mo>!</mo>
</mrow>
<mn>2</mn>
</mfrac>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>accept </mtext>
</mrow>
<mi>sin</mi>
<mo></mo>
<mi>x</mi>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>21</mn>
<mo>!</mo>
</mrow>
<mrow>
<mn>1</mn>
<mo>!</mo>
</mrow>
</mfrac>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>cos</mi>
<mo></mo>
<mi>x</mi>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>21</mn>
<mo>!</mo>
</mrow>
<mrow>
<mn>2</mn>
<mo>!</mo>
</mrow>
</mfrac>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>A1 N3</em></strong></p>
<p>(ii) substituting <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> (seen anywhere) <strong><em>(A1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f^{(19)}}(\pi ){g^{(20)}}(\pi ) + {f^{(20)}}(\pi ){g^{(19)}}(\pi ),{\text{ }}\sin \pi \frac{{21!}}{{1!}}\pi + \cos \pi \frac{{21!}}{{2!}}{\pi ^2}">
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mo stretchy="false">(</mo>
<mn>19</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
<mo stretchy="false">(</mo>
<mi>π</mi>
<mo stretchy="false">)</mo>
<mrow>
<msup>
<mi>g</mi>
<mrow>
<mo stretchy="false">(</mo>
<mn>20</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
<mo stretchy="false">(</mo>
<mi>π</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mo stretchy="false">(</mo>
<mn>20</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
<mo stretchy="false">(</mo>
<mi>π</mi>
<mo stretchy="false">)</mo>
<mrow>
<msup>
<mi>g</mi>
<mrow>
<mo stretchy="false">(</mo>
<mn>19</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
<mo stretchy="false">(</mo>
<mi>π</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>sin</mi>
<mo></mo>
<mi>π</mi>
<mfrac>
<mrow>
<mn>21</mn>
<mo>!</mo>
</mrow>
<mrow>
<mn>1</mn>
<mo>!</mo>
</mrow>
</mfrac>
<mi>π</mi>
<mo>+</mo>
<mi>cos</mi>
<mo></mo>
<mi>π</mi>
<mfrac>
<mrow>
<mn>21</mn>
<mo>!</mo>
</mrow>
<mrow>
<mn>2</mn>
<mo>!</mo>
</mrow>
</mfrac>
<mrow>
<msup>
<mi>π</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span></p>
<p>evidence of one correct value for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin \pi ">
<mi>sin</mi>
<mo></mo>
<mi>π</mi>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos \pi ">
<mi>cos</mi>
<mo></mo>
<mi>π</mi>
</math></span> (seen anywhere) <strong><em>(A1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin \pi = 0,{\text{ }}\cos \pi = - 1">
<mi>sin</mi>
<mo></mo>
<mi>π</mi>
<mo>=</mo>
<mn>0</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>cos</mi>
<mo></mo>
<mi>π</mi>
<mo>=</mo>
<mo>−</mo>
<mn>1</mn>
</math></span></p>
<p>evidence of correct values substituted into <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h'(\pi )">
<msup>
<mi>h</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>π</mi>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="21!(\pi )\left( {0 - \frac{\pi }{{2!}}} \right),{\text{ }}21!(\pi )\left( { - \frac{\pi }{2}} \right),{\text{ }}0 + ( - 1)\frac{{21!}}{2}{\pi ^2}">
<mn>21</mn>
<mo>!</mo>
<mo stretchy="false">(</mo>
<mi>π</mi>
<mo stretchy="false">)</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>0</mn>
<mo>−</mo>
<mfrac>
<mi>π</mi>
<mrow>
<mn>2</mn>
<mo>!</mo>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>21</mn>
<mo>!</mo>
<mo stretchy="false">(</mo>
<mi>π</mi>
<mo stretchy="false">)</mo>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>0</mn>
<mo>+</mo>
<mo stretchy="false">(</mo>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mfrac>
<mrow>
<mn>21</mn>
<mo>!</mo>
</mrow>
<mn>2</mn>
</mfrac>
<mrow>
<msup>
<mi>π</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span></p>
<p> </p>
<p><strong>Note: </strong>If candidates write only the first line followed by the answer, award <strong><em>A1A0A0</em></strong>.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{ - 21!}}{2}{\pi ^2}">
<mfrac>
<mrow>
<mo>−</mo>
<mn>21</mn>
<mo>!</mo>
</mrow>
<mn>2</mn>
</mfrac>
<mrow>
<msup>
<mi>π</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> <strong><em>AG N0</em></strong></p>
<p><strong><em>[7 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>A particle P starts from point O and moves along a straight line. The graph of its velocity, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v">
<mi>v</mi>
</math></span> ms<sup>−1</sup> after <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> seconds, for 0 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> ≤ 6 , is shown in the following diagram.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p>The graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v">
<mi>v</mi>
</math></span> has <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span>-intercepts when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> = 0, 2 and 4.</p>
<p>The function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s\left( t \right)">
<mi>s</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> represents the displacement of P from O after <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> seconds.</p>
<p>It is known that P travels a distance of 15 metres in the first 2 seconds. It is also known that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s\left( 2 \right) = s\left( 5 \right)">
<mi>s</mi>
<mrow>
<mo>(</mo>
<mn>2</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>s</mi>
<mrow>
<mo>(</mo>
<mn>5</mn>
<mo>)</mo>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_2^4 {v\,{\text{d}}t} = 9">
<msubsup>
<mo>∫<!-- ∫ --></mo>
<mn>2</mn>
<mn>4</mn>
</msubsup>
<mrow>
<mi>v</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
<mo>=</mo>
<mn>9</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s\left( 4 \right) - s\left( 2 \right)"> <mi>s</mi> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> <mo>−</mo> <mi>s</mi> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the total distance travelled in the first 5 seconds.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p>recognizing relationship between <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v"> <mi>v</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s"> <mi>s</mi> </math></span> <em><strong>(M1)</strong></em></p>
<p><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {v = s} "> <mo>∫</mo> <mrow> <mi>v</mi> <mo>=</mo> <mi>s</mi> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s' = v"> <msup> <mi>s</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>v</mi> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s\left( 4 \right) - s\left( 2 \right) = 9"> <mi>s</mi> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> <mo>−</mo> <mi>s</mi> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> <mo>=</mo> <mn>9</mn> </math></span> <em><strong>A1 N2</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>correctly interpreting distance travelled in first 2 seconds (seen anywhere, including part (a) or the area of 15 indicated on diagram) <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^2 {v = 15} "> <msubsup> <mo>∫</mo> <mn>0</mn> <mn>2</mn> </msubsup> <mrow> <mi>v</mi> <mo>=</mo> <mn>15</mn> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s\left( 2 \right) = 15"> <mi>s</mi> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> <mo>=</mo> <mn>15</mn> </math></span></p>
<p>valid approach to find total distance travelled <em><strong>(M1)</strong></em></p>
<p><em>eg</em> sum of 3 areas, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^4 {v + } \int_4^5 v "> <msubsup> <mo>∫</mo> <mn>0</mn> <mn>4</mn> </msubsup> <mrow> <mi>v</mi> <mo>+</mo> </mrow> <msubsup> <mo>∫</mo> <mn>4</mn> <mn>5</mn> </msubsup> <mi>v</mi> </math></span>, shaded areas in diagram between 0 and 5</p>
<p><strong>Note:</strong> Award <em><strong>M0</strong> </em>if only <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^5 {\left| v \right|} "> <msubsup> <mo>∫</mo> <mn>0</mn> <mn>5</mn> </msubsup> <mrow> <mrow> <mo>|</mo> <mi>v</mi> <mo>|</mo> </mrow> </mrow> </math></span> is seen.</p>
<p>correct working towards finding distance travelled between 2 and 5 (seen anywhere including within total area expression or on diagram) <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_2^4 {v - } \int_4^5 v "> <msubsup> <mo>∫</mo> <mn>2</mn> <mn>4</mn> </msubsup> <mrow> <mi>v</mi> <mo>−</mo> </mrow> <msubsup> <mo>∫</mo> <mn>4</mn> <mn>5</mn> </msubsup> <mi>v</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_2^4 {v = } \int_4^5 {\left| v \right|} "> <msubsup> <mo>∫</mo> <mn>2</mn> <mn>4</mn> </msubsup> <mrow> <mi>v</mi> <mo>=</mo> </mrow> <msubsup> <mo>∫</mo> <mn>4</mn> <mn>5</mn> </msubsup> <mrow> <mrow> <mo>|</mo> <mi>v</mi> <mo>|</mo> </mrow> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_4^5 {v\,{\text{d}}t} = - 9"> <msubsup> <mo>∫</mo> <mn>4</mn> <mn>5</mn> </msubsup> <mrow> <mi>v</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> <mo>=</mo> <mo>−</mo> <mn>9</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s\left( 4 \right) - s\left( 2 \right) - \left[ {s\left( 5 \right) - s\left( 4 \right)} \right]"> <mi>s</mi> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> <mo>−</mo> <mi>s</mi> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> <mo>−</mo> <mrow> <mo>[</mo> <mrow> <mi>s</mi> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> <mo>−</mo> <mi>s</mi> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> </math></span>,</p>
<p>equal areas <img src="data:image/png;base64,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"></p>
<p>correct working using <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s\left( 5 \right) = s\left( 2 \right)"> <mi>s</mi> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> <mo>=</mo> <mi>s</mi> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </math></span> <em><strong>(A1)</strong></em></p>
<p><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="15 + 9 - \left( { - 9} \right)"> <mn>15</mn> <mo>+</mo> <mn>9</mn> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>9</mn> </mrow> <mo>)</mo> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="15 + 2\left[ {s\left( 4 \right) - s\left( 2 \right)} \right]"> <mn>15</mn> <mo>+</mo> <mn>2</mn> <mrow> <mo>[</mo> <mrow> <mi>s</mi> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> <mo>−</mo> <mi>s</mi> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="15 + 2\left( 9 \right)"> <mn>15</mn> <mo>+</mo> <mn>2</mn> <mrow> <mo>(</mo> <mn>9</mn> <mo>)</mo> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2 \times s\left( 4 \right) - s\left( 2 \right)"> <mn>2</mn> <mo>×</mo> <mi>s</mi> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> <mo>−</mo> <mi>s</mi> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="48 - 15"> <mn>48</mn> <mo>−</mo> <mn>15</mn> </math></span></p>
<p>total distance travelled = 33 (m) <em><strong>A1 N2</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = 15 - {x^2}"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>15</mn> <mo>−</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </math></span>, for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \in \mathbb{R}"> <mi>x</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> </math></span>. The following diagram shows part of the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span> and the rectangle OABC, where A is on the negative <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-axis, B is on the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span>, and C is on the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span>-axis.</p>
<p><img src="images/Schermafbeelding_2018-02-11_om_13.13.04.png" alt="N17/5/MATME/SP1/ENG/TZ0/06"></p>
<p>Find the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-coordinate of A that gives the maximum area of OABC.</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 find the area of OABC <strong><em>(M1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{OA}} \times {\text{OC, }}x \times f(x),{\text{ }}f(x) \times ( - x)"> <mrow> <mtext>OA</mtext> </mrow> <mo>×</mo> <mrow> <mtext>OC, </mtext> </mrow> <mi>x</mi> <mo>×</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span></p>
<p>correct expression for area in one variable <strong><em>(A1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{area}} = x(15 - {x^2}),{\text{ }}15x - {x^3},{\text{ }}{x^3} - 15x"> <mrow> <mtext>area</mtext> </mrow> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mn>15</mn> <mo>−</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>15</mn> <mi>x</mi> <mo>−</mo> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> <mo>−</mo> <mn>15</mn> <mi>x</mi> </math></span></p>
<p>valid approach to find maximum <strong>area</strong> (seen anywhere) <strong><em>(M1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A’(x) = 0"> <msup> <mi>A</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </math></span></p>
<p>correct derivative <strong><em>A1</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="15 - 3{x^2},{\text{ }}(15 - {x^2}) + x( - 2x) = 0,{\text{ }} - 15 + 3{x^2}"> <mn>15</mn> <mo>−</mo> <mn>3</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mo stretchy="false">(</mo> <mn>15</mn> <mo>−</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mo>−</mo> <mn>15</mn> <mo>+</mo> <mn>3</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </math></span></p>
<p>correct working <strong><em>(A1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="15 = 3{x^2},{\text{ }}{x^2} = 5,{\text{ }}x = \sqrt 5 "> <mn>15</mn> <mo>=</mo> <mn>3</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mn>5</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>x</mi> <mo>=</mo> <msqrt> <mn>5</mn> </msqrt> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = - \sqrt 5 {\text{ }}\left( {{\text{accept A}}\left( { - \sqrt 5 ,{\text{ }}0} \right)} \right)"> <mi>x</mi> <mo>=</mo> <mo>−</mo> <msqrt> <mn>5</mn> </msqrt> <mrow> <mtext> </mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>accept A</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <msqrt> <mn>5</mn> </msqrt> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </math></span> <strong><em>A2 N3</em></strong></p>
<p><strong><em>[7 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>Consider a function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span>. The line <em>L</em><sub>1</sub> with equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 3x + 1">
<mi>y</mi>
<mo>=</mo>
<mn>3</mn>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</math></span> is a tangent to the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 2">
<mi>x</mi>
<mo>=</mo>
<mn>2</mn>
</math></span></p>
</div>
<div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( x \right) = f\left( {{x^2} + 1} \right)">
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and P be the point on the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g">
<mi>g</mi>
</math></span> where <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>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( 2 \right)"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </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>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( 2 \right)"> <mi>f</mi> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the graph of <em>g</em> has a gradient of 6 at 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>Let <em>L</em><sub>2</sub> be the tangent to the graph of <em>g</em> at P. <em>L</em><sub>1</sub> intersects <em>L</em><sub>2</sub> at the point Q.</p>
<p>Find the y-coordinate of Q.</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>recognize that <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> is the gradient of the tangent at <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><em>eg </em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right) = m"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>m</mi> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( 2 \right) = 3"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> <mo>=</mo> <mn>3</mn> </math></span> (accept <em>m</em> = 3) <em><strong>A1 N2</strong></em></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>recognize that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( 2 \right) = y\left( 2 \right)"> <mi>f</mi> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> <mo>=</mo> <mi>y</mi> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </math></span> <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( 2 \right) = 3 \times 2 + 1"> <mi>f</mi> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> <mo>=</mo> <mn>3</mn> <mo>×</mo> <mn>2</mn> <mo>+</mo> <mn>1</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( 2 \right) = 7"> <mi>f</mi> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> <mo>=</mo> <mn>7</mn> </math></span> <em><strong>A1 N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognize that the gradient of the graph of <em>g</em> is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'\left( x \right)"> <msup> <mi>g</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span> <em><strong>(M1)</strong></em></p>
<p>choosing chain rule to find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'\left( x \right)"> <msup> <mi>g</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span> <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}u}} \times \frac{{{\text{d}}u}}{{{\text{d}}x}},\,\,u = {x^2} + 1,\,\,u' = 2x"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>u</mi> </mrow> </mfrac> <mo>×</mo> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>u</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mi>u</mi> <mo>=</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <msup> <mi>u</mi> <mo>′</mo> </msup> <mo>=</mo> <mn>2</mn> <mi>x</mi> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'\left( x \right) = f'\left( {{x^2} + 1} \right) \times 2x"> <msup> <mi>g</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>×</mo> <mn>2</mn> <mi>x</mi> </math></span> <em><strong>A2</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'\left( 1 \right) = 3 \times 2"> <msup> <mi>g</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <mn>3</mn> <mo>×</mo> <mn>2</mn> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'\left( 1 \right) = 6"> <msup> <mi>g</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <mn>6</mn> </math></span> <em><strong>AG N0 </strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
<p> </p>
<p> </p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p> at Q, <em>L</em><sub>1</sub> = <em>L</em><sub>2</sub> (seen anywhere) <em><strong> (M1)</strong></em></p>
<p>recognize that the gradient of <em>L</em><sub>2</sub> is <em>g'</em>(1) (seen anywhere) <em><strong> (M1)</strong></em><br><em>eg</em> <em>m</em> = 6</p>
<p>finding <em>g </em>(1) (seen anywhere) <em><strong>(A1)</strong></em><br><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( 1 \right) = f\left( 2 \right),\,\,g\left( 1 \right) = 7"> <mi>g</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mi>g</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <mn>7</mn> </math></span></p>
<p>attempt to substitute gradient and/or coordinates into equation of a straight line <em><strong>M1</strong></em><br><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y - g\left( 1 \right) = 6\left( {x - 1} \right),\,\,y - 1 = g'\left( 1 \right)\left( {x - 7} \right),\,\,7 = 6\left( 1 \right) + {\text{b}}"> <mi>y</mi> <mo>−</mo> <mi>g</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <mn>6</mn> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mi>y</mi> <mo>−</mo> <mn>1</mn> <mo>=</mo> <msup> <mi>g</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mn>7</mn> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mn>7</mn> <mo>=</mo> <mn>6</mn> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mtext>b</mtext> </mrow> </math></span></p>
<p>correct equation for <em>L</em><sub>2</sub> </p>
<p><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y - 7 = 6\left( {x - 1} \right),\,\,y = 6x + 1"> <mi>y</mi> <mo>−</mo> <mn>7</mn> <mo>=</mo> <mn>6</mn> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mi>y</mi> <mo>=</mo> <mn>6</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> </math></span> <em><strong>A1</strong></em></p>
<p>correct working to find Q <em><strong>(A1)</strong></em><br><em>eg </em>same <em>y</em>-intercept, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3x = 0"> <mn>3</mn> <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="y = 1"> <mi>y</mi> <mo>=</mo> <mn>1</mn> </math></span> <em><strong>A1 N2</strong></em></p>
<p><em><strong>[7 marks]</strong></em></p>
<p> </p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>A closed cylindrical can with radius r centimetres and height h centimetres has a volume of 20<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi ">
<mi>π<!-- π --></mi>
</math></span> cm<sup>3</sup>.</p>
<p style="text-align: center;"><img 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"></p>
</div>
<div class="specification">
<p>The material for the base and top of the can costs 10 cents per cm<sup>2</sup> and the material for the curved side costs 8 cents per cm<sup>2</sup>. The total cost of the material, in cents, is <em>C</em>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Express <em>h</em> in terms of <em>r.</em></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 <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C = 20\pi {r^2} + \frac{{320\pi }}{r}">
<mi>C</mi>
<mo>=</mo>
<mn>20</mn>
<mi>π</mi>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mfrac>
<mrow>
<mn>320</mn>
<mi>π</mi>
</mrow>
<mi>r</mi>
</mfrac>
</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>Given that there is a minimum value for <em>C</em>, find this minimum value in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi ">
<mi>π</mi>
</math></span>.</p>
<div class="marks">[9]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>correct equation for volume <strong><em>(A1)</em></strong><br><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi {r^2}h = 20\pi ">
<mi>π</mi>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
<mi>h</mi>
<mo>=</mo>
<mn>20</mn>
<mi>π</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h = \frac{{20}}{{{r^2}}}">
<mi>h</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>20</mn>
</mrow>
<mrow>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span> <strong><em>A1 N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<p> </p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to find formula for cost of parts <em><strong> (M1)</strong></em><br><em>eg </em> 10 × two circles, 8 × curved side</p>
<p>correct expression for cost of two circles in terms of <em>r</em> (seen anywhere) <em><strong>A1</strong></em><br><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\pi {r^2} \times 10">
<mn>2</mn>
<mi>π</mi>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>×</mo>
<mn>10</mn>
</math></span></p>
<p>correct expression for cost of curved side (seen anywhere) <em><strong>(A1)</strong></em><br><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\pi r \times h \times 8">
<mn>2</mn>
<mi>π</mi>
<mi>r</mi>
<mo>×</mo>
<mi>h</mi>
<mo>×</mo>
<mn>8</mn>
</math></span></p>
<p>correct expression for cost of curved side in terms of <em>r </em> <em><strong>A1</strong></em><br>eg <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="8 \times 2\pi r \times \frac{{20}}{{{r^2}}},\,\,\frac{{320\pi }}{{{r^2}}}">
<mn>8</mn>
<mo>×</mo>
<mn>2</mn>
<mi>π</mi>
<mi>r</mi>
<mo>×</mo>
<mfrac>
<mrow>
<mn>20</mn>
</mrow>
<mrow>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mfrac>
<mrow>
<mn>320</mn>
<mi>π</mi>
</mrow>
<mrow>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C = 20\pi {r^2} + \frac{{320\pi }}{r}">
<mi>C</mi>
<mo>=</mo>
<mn>20</mn>
<mi>π</mi>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mfrac>
<mrow>
<mn>320</mn>
<mi>π</mi>
</mrow>
<mi>r</mi>
</mfrac>
</math></span> <em><strong>AG N0</strong></em></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>recognize <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C' = 0">
<msup>
<mi>C</mi>
<mo>′</mo>
</msup>
<mo>=</mo>
<mn>0</mn>
</math></span> at minimum <em><strong>(R1)</strong></em><br>eg <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C' = 0,\,\,\frac{{{\text{d}}C}}{{{\text{d}}r}} = 0">
<msup>
<mi>C</mi>
<mo>′</mo>
</msup>
<mo>=</mo>
<mn>0</mn>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>C</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>r</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p>correct differentiation (may be seen in equation)</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C' = 40\pi r - \frac{{320\pi }}{{{r^2}}}">
<msup>
<mi>C</mi>
<mo>′</mo>
</msup>
<mo>=</mo>
<mn>40</mn>
<mi>π</mi>
<mi>r</mi>
<mo>−</mo>
<mfrac>
<mrow>
<mn>320</mn>
<mi>π</mi>
</mrow>
<mrow>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span> <em><strong>A1A1</strong></em></p>
<p>correct equation <em><strong>A1</strong></em><br>eg <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="40\pi r - \frac{{320\pi }}{{{r^2}}} = 0,\,\,40\pi r\frac{{320\pi }}{{{r^2}}}">
<mn>40</mn>
<mi>π</mi>
<mi>r</mi>
<mo>−</mo>
<mfrac>
<mrow>
<mn>320</mn>
<mi>π</mi>
</mrow>
<mrow>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0</mn>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mn>40</mn>
<mi>π</mi>
<mi>r</mi>
<mfrac>
<mrow>
<mn>320</mn>
<mi>π</mi>
</mrow>
<mrow>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span></p>
<p>correct working <em><strong>(A1)</strong></em><br>eg <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="40{r^3} = 320,\,\,{r^3} = 8">
<mn>40</mn>
<mrow>
<msup>
<mi>r</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>320</mn>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<msup>
<mi>r</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>8</mn>
</math></span></p>
<p><em>r</em> = 2 (m) <em><strong>A1</strong></em></p>
<p>attempt to substitute <strong>their</strong> value of <em>r</em> into <em>C</em><br>eg <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="20\pi \times 4 + 320 \times \frac{\pi }{2}">
<mn>20</mn>
<mi>π</mi>
<mo>×</mo>
<mn>4</mn>
<mo>+</mo>
<mn>320</mn>
<mo>×</mo>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</math></span> <em><strong>(M1)</strong></em></p>
<p>correct working<br>eg <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="80\pi + 160\pi ">
<mn>80</mn>
<mi>π</mi>
<mo>+</mo>
<mn>160</mn>
<mi>π</mi>
</math></span> <em><strong> (A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="240\pi ">
<mn>240</mn>
<mi>π</mi>
</math></span> (cents) <em><strong>A1 N3</strong></em></p>
<p><strong>Note:</strong> Do not accept 753.6, 753.98 or 754, even if 240<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi ">
<mi>π</mi>
</math></span> is seen.</p>
<p><em><strong>[9 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'(x) = {\sin ^3}(2x)\cos (2x)">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<msup>
<mi>sin</mi>
<mn>3</mn>
</msup>
</mrow>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mi>cos</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span>. Find <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>, given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( {\frac{\pi }{4}} \right) = 1">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>1</mn>
</math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>evidence of integration <strong><em>(M1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {f'(x){\text{d}}x} ">
<mo>∫</mo>
<mrow>
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</math></span></p>
<p>correct integration (accept missing <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C">
<mi>C</mi>
</math></span>) <strong><em>(A2)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2} \times \frac{{{{\sin }^4}(2x)}}{4},{\text{ }}\frac{1}{8}{\sin ^4}(2x) + C">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mi>sin</mi>
</mrow>
<mn>4</mn>
</msup>
</mrow>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mrow>
<mn>4</mn>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mfrac>
<mn>1</mn>
<mn>8</mn>
</mfrac>
<mrow>
<msup>
<mi>sin</mi>
<mn>4</mn>
</msup>
</mrow>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mi>C</mi>
</math></span></p>
<p>substituting initial condition into their integrated expression (must have <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" + C">
<mo>+</mo>
<mi>C</mi>
</math></span>) <strong><em>M1</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 = \frac{1}{8}{\sin ^4}\left( {\frac{\pi }{2}} \right) + C">
<mn>1</mn>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>8</mn>
</mfrac>
<mrow>
<msup>
<mi>sin</mi>
<mn>4</mn>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>C</mi>
</math></span></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>M0 </em></strong>if they substitute into the original or differentiated function.</p>
<p> </p>
<p>recognizing <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin \left( {\frac{\pi }{2}} \right) = 1">
<mi>sin</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>1</mn>
</math></span> <strong><em>(A1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 = \frac{1}{8}{(1)^4} + C">
<mn>1</mn>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>8</mn>
</mfrac>
<mrow>
<mo stretchy="false">(</mo>
<mn>1</mn>
<msup>
<mo stretchy="false">)</mo>
<mn>4</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>C</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C = \frac{7}{8}">
<mi>C</mi>
<mo>=</mo>
<mfrac>
<mn>7</mn>
<mn>8</mn>
</mfrac>
</math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = \frac{1}{8}{\sin ^4}(2x) + \frac{7}{8}">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>8</mn>
</mfrac>
<mrow>
<msup>
<mi>sin</mi>
<mn>4</mn>
</msup>
</mrow>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mfrac>
<mn>7</mn>
<mn>8</mn>
</mfrac>
</math></span> <strong><em>A1 N5</em></strong></p>
<p><strong><em>[7 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p>Consider <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = \log k(6x - 3{x^2})">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>log</mi>
<mo></mo>
<mi>k</mi>
<mo stretchy="false">(</mo>
<mn>6</mn>
<mi>x</mi>
<mo>−</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</math></span>, for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 < x < 2">
<mn>0</mn>
<mo><</mo>
<mi>x</mi>
<mo><</mo>
<mn>2</mn>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k > 0">
<mi>k</mi>
<mo>></mo>
<mn>0</mn>
</math></span>.</p>
<p>The equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = 2">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>2</mn>
</math></span> has exactly one solution. Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><strong>METHOD 1 – using discriminant</strong></p>
<p>correct equation without logs <strong><em>(A1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="6x - 3{x^2} = {k^2}">
<mn>6</mn>
<mi>x</mi>
<mo>−</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mi>k</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span></p>
<p>valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 3{x^2} + 6x - {k^2} = 0,{\text{ }}3{x^2} - 6x + {k^2} = 0">
<mo>−</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>6</mn>
<mi>x</mi>
<mo>−</mo>
<mrow>
<msup>
<mi>k</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>0</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>6</mn>
<mi>x</mi>
<mo>+</mo>
<mrow>
<msup>
<mi>k</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p>recognizing discriminant must be zero (seen anywhere) <strong><em>M1</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\Delta = 0">
<mi mathvariant="normal">Δ</mi>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p>correct discriminant <strong><em>(A1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{6^2} - 4( - 3)( - {k^2}),{\text{ }}36 - 12{k^2} = 0">
<mrow>
<msup>
<mn>6</mn>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>4</mn>
<mo stretchy="false">(</mo>
<mo>−</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mo>−</mo>
<mrow>
<msup>
<mi>k</mi>
<mn>2</mn>
</msup>
</mrow>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>36</mn>
<mo>−</mo>
<mn>12</mn>
<mrow>
<msup>
<mi>k</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p>correct working <strong><em>(A1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="12{k^2} = 36,{\text{ }}{k^2} = 3">
<mn>12</mn>
<mrow>
<msup>
<mi>k</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>36</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<msup>
<mi>k</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>3</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = \sqrt 3 ">
<mi>k</mi>
<mo>=</mo>
<msqrt>
<mn>3</mn>
</msqrt>
</math></span> <strong><em>A2 N2</em></strong></p>
<p><strong>METHOD 2 – completing the square</strong></p>
<p>correct equation without logs <strong><em>(A1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="6x - 3{x^2} = {k^2}">
<mn>6</mn>
<mi>x</mi>
<mo>−</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mi>k</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span></p>
<p>valid approach to complete the square <strong><em>(M1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3({x^2} - 2x + 1) = - {k^2} + 3,{\text{ }}{x^2} - 2x + 1 - 1 + \frac{{{k^2}}}{3} = 0">
<mn>3</mn>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>−</mo>
<mrow>
<msup>
<mi>k</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>3</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
<mo>−</mo>
<mn>1</mn>
<mo>+</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>k</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mn>3</mn>
</mfrac>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p>correct working <strong><em>(A1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3{(x - 1)^2} = - {k^2} + 3,{\text{ }}{(x - 1)^2} - 1 + \frac{{{k^2}}}{3} = 0">
<mn>3</mn>
<mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
<msup>
<mo stretchy="false">)</mo>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mo>−</mo>
<mrow>
<msup>
<mi>k</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>3</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
<msup>
<mo stretchy="false">)</mo>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
<mo>+</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>k</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mn>3</mn>
</mfrac>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p>recognizing conditions for one solution <strong><em>M1</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{(x - 1)^2} = 0,{\text{ }} - 1 + \frac{{{k^2}}}{3} = 0">
<mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
<msup>
<mo stretchy="false">)</mo>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>0</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo>−</mo>
<mn>1</mn>
<mo>+</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>k</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mn>3</mn>
</mfrac>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p>correct working <strong><em>(A1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{k^2}}}{3} = 1,{\text{ }}{k^2} = 3">
<mfrac>
<mrow>
<mrow>
<msup>
<mi>k</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mn>3</mn>
</mfrac>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<msup>
<mi>k</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>3</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = \sqrt 3 ">
<mi>k</mi>
<mo>=</mo>
<msqrt>
<mn>3</mn>
</msqrt>
</math></span> <strong> <em>A2 N2</em></strong></p>
<p><strong><em>[7 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = \frac{{6 - 2x}}{{\sqrt {16 + 6x - {x^2}} }}">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>6</mn>
<mo>−</mo>
<mn>2</mn>
<mi>x</mi>
</mrow>
<mrow>
<msqrt>
<mn>16</mn>
<mo>+</mo>
<mn>6</mn>
<mi>x</mi>
<mo>−</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mrow>
</mfrac>
</math></span>. The following diagram shows part of the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span>.</p>
<p style="text-align: center;"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAYYAAAFRCAYAAABwuwpyAAAgAElEQVR4Ae3dDXBV5ZnA8ScuyxpFjVArIEgVl+CslATBL0KBWghaVimsdmiB7EhAsKHUDz4U1LrbUArYrOAHKx9bEWWVktG1IqQdB2wYWxASwamb7LYOAhHHGtjW3bDWcneeIyd7AzfJvfecc9/3nPO/M07CvefjfX/Phcfnfc95T14ikUgILwQQQAABBE4JnIUEAggggAACyQIkhmQNfkcAAQQQEBIDXwIEEEAAgTYCJIY2HPwBAQQQQIDEwHcAAQQQQKCNAImhDQd/QAABBBAgMfAdQAABBBBoI0BiaMPBHxBAAAEESAwdfQc+qJayvDzJ0/9GV8neT05+vvUfXpcFvQfJ9OqDcuqdjo7CZwgggECoBEgMHYWr10R5JtEiDetuE9nxprzd9OnnW3crlNK/7yeHj7d0tDefIYAAAqEUIDF0Graz5fLB10h/+Uia//jZ51uf1UP6XPE3MmZwT24E6dSPDRBAIGwCJIY0InbWed2lvxySAwePOVufPLJVfrz7WplcXJDG3myCAAIIhEuAxJBGvM46r0B6tm53XOpebJSxD46XS9BrVeEXBBCIjgD/tKUTy3MLpGev38qu370vh15fK1v6TZYJl3RNZ0+2QQABBEInQGJIJ2TnXCAXnSPyP3Xr5Z92DJY5E/oxt5COG9sggEAoBUgM6YTtrHOle/9eIl2+IjPm3yi9UEtHjW0QQCCkAvwTl27g+nxHNi6dLAO7QZYuGdshgEA4BfhXrtO4HZe9j70gMm+OfLUX8wqdcrEBAgiEXoDEkDKEx2XvivHS+971sr3qx/LGV+bKHQPPP2PL5uZmWbJkiRw+fPiMz3gDAQQQCKsAiSFl5D6T//roqHywr0GOfmW2zL069f0Kq1evlkWLFskjjzyS8ii8iQACCIRRIC+RSCTC2HDTba6vr5fi4uLWZmzfvl3Gjh3b+md+QQABBMIqQMWQReRaWlpk5syZUl5e7uxdWVkpixcvFh1a4oUAAgiEXYDEkEUEN2/eLHv27JGHH37Y2XvWrFnOz6VLl2ZxNHZBAAEE7BJgKCnDeDQ2NkphYaFs2bJFJk6c6CzJraNxNTU1UlpaKnV1dVJUVJThUdkcAQQQsEeAxJBhLCZMmODssWnTJsnPz29NDPrm/PnzZceOHbJz507nswwPzeYIIICAFQIkhgzCUF1dLZMmTZKGhgYZMGCAs6c+xMedv9fLVvv27SurVq2SioqKDI7MpggggIA9AiSGNGOhE8s9evQ44x/95MSgh0qVPNI8BZshgAACVggw+ZxBGLQSmD59eod76LyDbscLAQQQCKsAFYPHyJ1eMXg8HLsjgAACxgWoGIyHgAYggAACdgmQGOyKB61BAAEEjAuQGIyHgAYggAACdgmQGOyKB61BAAEEjAuQGIyHgAYggAACdgmQGOyKB61BAAEEjAuQGIyHgAYggAACdgmQGOyKB61BAAEEjAuQGIyHgAYggAACdgmQGOyKB61BAAEEjAuQGIyHgAYggAACdgmQGOyKB61BAAEEjAuQGIyHgAYggAACdgmQGOyKB61BAAEEjAuQGIyHgAYggAACdgmQGOyKB61BAAEEjAuQGIyHgAYggAACdgmQGOyKB61BAAEEjAuQGIyHgAYggAACdgmQGOyKB61BAAEEjAuQGIyHgAYggAACdgmQGOyKB61BAAEEjAuQGIyHgAYggAACdgmQGOyKB61BAAEEjAuQGIyHgAYggAACdgmQGOyKB61BAAEEjAuQGIyHgAYggAACdgmQGOyKB61BAAEEjAuQGIyHgAYggAACdgmQGOyKB61BAAEEjAuQGIyHgAYggAACdgnkPDG0tLRIY2OjXQq0BgEEEECgVSDnieGXv/ylTJkypbUB/IIAAgggYJdAXiKRSOSySc3NzdKjRw+pq6uToqKiXJ46kHPl5eVJjgkD6QcHRQABBFyBnFcM3bt3l8rKStm6davbBn4igAACCFgkkPOKQfteX18vxcXF8vHHH4smijC/qBjCHD3ajgACqQRyXjFoI3QI6dZbb5UdO3akahPvIYAAAggYFDCSGLS/06ZNkw0bNhjsOqdGAAEEEEglYGQoSRviTkLX1tbK8OHDU7UtFO8xlBSKMNFIBBDIQMBYxeBOQr/88ssZNJdNEUAAAQSCFjBWMWjH9Ea3wsLCUE9CUzEE/RXl+AggkGsBYxWDdnTAgAHOJPTzzz+f635zPgQQQACBdgSMJgZt01133eVMQutSGbwQQAABBMwLGE8MI0aMcBT27dtnXoMWIIAAAgiI8cSQn5/vXLq6fPlywoEAAgggYIGA0clnt//upasNDQ3OvIP7fhh+MvkchijRRgQQyETAeMWgjdVLV+fNmydr167NpO1siwACCCAQgIAVFYP2K6zrJ1ExBPCt5JAIIGBUwIqKQQXc9ZO4dNXo94GTI4AAAuYnn5NjoMNJun4Sl64mq/A7AgggkFsBayoG7faQIUOc3r/22mu5VeBsCCCAAAKtAlYlBr10deHChbJ06dLWBvILAggggEBuBayZfHa77V66GpZVV5l8diPHTwQQiIqAVRWDouqlq6tWrRJueIvKV4x+IIBA2ASsqxgU0K0a6urqnKuVbEalYrA5OrQNAQSyEbCuYtBOuM9q4NLVbELKPggggIA3ASsrBu2S+6wG25fJoGLw9gVkbwQQsE/A2sSgVPPnz3fEli1bZp/cqRaRGKwNDQ1DAIEsBaxODGGoGkgMWX7z2A0BBKwVsHKOwdXSJ7zp3dA1NTXuW/xEAAEEEAhYwOqKQfu+a9cuKSkpsfa50FQMAX9DOTwCCORcwOqKQTWGDx/Oc6Fz/rXghAggEGcB6ysGDY7NVQMVQ5z/+tB3BKIpYH3FoOxUDdH88tErBBCwUyAUFYPS2Vo1UDHY+cWmVQggkL1AKCoG7R5VQ/ZBZk8EEEAgE4HQVAzaKRurBiqGTL5ubIsAAmEQsKxi+FQ+2PusLBjdW/LyesvoFbvlkyRFqoYkDH5FAAEEAhKwKDF8Kkeq75Gr/3aHFD5VLw3rSmTHz9+RppNte643vM2ZM8dZgbXtJ/wJAQQQQMAPAWuGkk4eqZYZw5bIFzZukx999Qsd9s2mNZQYSuowVHyIAAIhFLAkMRyXvSumyNCfT5SG1+6QAZ3UMTatoURiCOG3niYjgECHAp38E9zhvj59eFI+2fsvct+8fTL2mzfIFWm0yF1Dae3atT61gcMggAACCLgChiuG38vrC8bJjcv2uu2R/svfkn+/72rp0vpO6l9sqRqoGFLHh3cRQCC8AoYTg8J9Jh9UV0jvSc2yrmGD3DHg7LQ1da7h2LFjsmbNmrT38XtDEoPfohwPAQRMC1iQGE5VDfV3pTW/kAzmPhu6trbWuQEu+bNc/U5iyJU050EAgVwJpDGiH3BTPjso+7bslV5FX5KeGbZGnw29atUqWb58ecCN5PAIIIBAfAQy/KfYf5iTv3tbfv7b/jLm2r+W87M4/Le+9S1pamqS6urqLPZmFwQQQACB0wUMJ4bP5MN3dkuNDJGRV110etvS+rNWDQsXLpSlS5dKS0tLWvuwEQIIIIBA+wKGE8NxeffXb4mMHSclV6Q/6Xx6d2666Sbnrc2bN5/+EX9GAAEEEMhQwGxicOYXmtK+f6G9vuXn50tVVZWUlZWxVEZ7SLyPAAIIpClgNDF8Pr9QIt8s+ZJ4bYgusFdeXu4MKaXZdzZDAAEEEEghYPBy1RPSuH663Nk8V1657xrplqJxmb5l4qY3LlfNNEpsjwACtgt4/R/1zPp38qBUTx8qpevfliO710nlCwPkwW8X+ZIUtCG6VEZlZaW4i+xl1ji2RgABBBBQgdwmhrMukH5X9pSa6UUy7AmR7/zkfvlqr66+RmLWrFny8ssvS01Nja/H5WAIIIBAXAQMDiUFR6z3NOjlqzt37hSdmA7yxVBSkLocGwEETAjktmLIUQ/18tXevXvLunXrcnRGToMAAghERyCSFYOGp76+XoqLi6WhocGZewgqZFQMQclyXAQQMCUQ2cSgoLlYfZXEYOqry3kRQCAogUgnBnf11e3bt8vYsWMDMSQxBMLKQRFAwKBAJOcYXE9dR2nLli2yePFi7oh2UfiJAAIIdCIQ6cSgfXcnolevXt0JBR8jgAACCKhApIeS3BC7d0TX1dVJUVGR+7YvPxlK8oWRgyCAgEUCsUgM6r1kyRLZvXu3bNq0ydd7G0gMFn2baQoCCPgiEPmhJFfp7rvvdh7ow9Lcrgg/EUAAgdQCsakYtPu6TEZpaakcOnRI+vTpk1okw3epGDIEY3MEELBeIFaJQaPh970NJAbrv+M0EAEEMhSIzVCS66KPAX377bd5RrQLwk8EEEDgNIHYVQzafz+HlKgYTvtG8UcEEAi9QCwTg0bNryElEkPo/w7QAQQQOE0gdkNJbv8ZUnIl+IkAAgi0FYhtxaAMfgwpUTG0/ULxJwQQCL9ArBODhs/rkBKJIfx/CegBAgi0FYjtUJLL4A4pbdiwwX2LnwgggECsBWJfMWj03SGlbB7qQ8UQ678/dB6BSAqQGE6FNdu1lEgMkfx7QacQiLVA7IeS3Oi7aylVVVW5b/ETAQQQiKUAFUNS2N3luWtra2X48OFJn7T/KxVD+zZ8ggAC4RSgYkiK24ABA+SZZ54RrR70saC8EEAAgTgKUDGkiPqMGTOcd9esWZPi07ZvUTG09eBPCCAQfgEqhhQxfPjhh52F9riENQUObyGAQOQFqBjaCfGuXbukpKREOnscKBVDO4C8jQACoRWgYmgndDr5vGrVKpk5cybzDe0Y8TYCCERTgIqhg7i2tLTId7/7XWeL9uYbqBg6AOQjBBAIpQAVQwdhy8/PF+YbOgDiIwQQiKQAFUMaYXXnG1Ld30DFkAYgmyCAQKgEqBjSCJfON7j3Nxw+fDiNPdgEAQQQCK8AFUMGsdP7Gz766CPZtGmT6DCTvqgYMgBkUwQQCIUAFUMGYVq5cqU0NTUJ6yllgMamCCAQOoEuoWuxwQZrlVBdXS19+/aVPn36yLRp0wy2hlMjgAACwQiQGDJ01YSgk9B681v//v0z3JvNEUAAAfsFmGPIMkaPP/646JIZe/bskUQikeVR2A0BBBCwT4A5hixjUlFRIaNGjXL21hvheCGAAAJREaBi8BBJTQjnnHOOM9ewevXq1iuVPBySXRFAAAHjAlQMHkLgXrKqN8AtW7bMw5HYFQEEELBHgMlnH2Ixd+5cZ02lgoIC0d95IYAAAmEWoGLwIXrdu3eXhx56SL73ve/J1q1bfTgih0AAAQTMCZAYfLLXx4Jqcvj6178uOrTECwEEEAirAInBx8hpcigrK3Pucaivr/fxyBwKAQQQyJ0Acww+W48ZM8Y54pQpU2Tbtm3OHdI+n4LDIYAAAoEKUDEEwKvJYejQoc6wEquxBgDMIRFAIFABEkNAvCNHjhSdlNY5B5JDQMgcFgEEAhEgMQTCKtK1a1fnxjeSQ0DAHBYBBAITIDEERktyCJCWQyOAQIACJIYAcfXQyZXDHXfcIayrFDA4h0cAAc8CJAbPhJ0fwE0Of/rTn4Tk0LkXWyCAgFkBEkOO/N3kcPToUSc5NDc35+jMnAYBBBDITIDEkJmXp62Tk8Po0aO5WsmTJjsjgEBQAiSGoGTbOa6bHLhaqR0g3kYAAeMCJAYDISA5GEDnlAggkLYAiSFtKn83JDn468nREEDAPwESg3+WGR+J5JAxGTsggEAOBEgMOUDu6BRuciguLpZx48YJq7J2pMVnCCCQCwESQy6UOzmHJgd34T1NEDzPoRMwPkYAgUAFWHY7UN7MDu4u2V1SUiK1tbUyfPjwzA7A1ggggIAPAlQMPiD6eQhNDvrcaE0O69ev9/PQHAsBBBBIS4DEkBZTbjcaNmyY85jQ6dOny2OPPZbbk3M2BBCIvQCJwdKvgPsM6SeffNJJEiy+Z2mgaBYCERQgMVgcVE0OFRUV8vzzz8usWbNYmdXiWNE0BKIkQGKwPJq6dIbOObz77rtSXl7O+kqWx4vmIRAFARJDCKKoyWH27NnS1NTEo0JDEC+aiEDYBUgMIYmgeyPcwIEDnRvhuNchJIGjmQiEUIDEEKKgaXK45ZZbZOjQoc7lrFu3bg1R62kqAgiERYDEEJZIJbVT73VYsGCBM6zE5axJMPyKAAK+CJAYfGHM/UEGDRoklZWV8sQTT0hZWRlXLOU+BJwRgcgKkBhCHNp+/frJnDlznLWVuGIpxIGk6QhYJkBisCwgmTZHr1hatGiRc8XSzTffLI2NjZkegu0RQACBNgIkhjYc4fyDe8XSkCFDpLCwUGpqasLZEVqNAAJWCJAYrAiD90a4S3frzXClpaWsseSdlCMgEFsBEkPEQq8L8DEpHbGg0h0EcixAYsgxeC5O505K6zIa1113Hcto5AKdcyAQIQESQ4SCmdwVdxmNvn37iv7HndLJOvyOAAIdCZAYOtIJ+Wc67zB58mS58847nTuluRku5AGl+QjkSIBHe+YI2uRpRowYIRdffLFzM9y+fftk9erVkp+fb7JJnBsBBCwWoGKwODh+Nk2f7aA3wzHv4Kcqx0IgmgIkhmjGNWWvTp934H6HlEy8iUDsBUgMMfsKuPMOyfc78NjQmH0J6C4CnQiQGDoBiurH7v0Oq1atkpkzZ3JJa1QDTb8QyEKAxJAFWlR20fsd5s2b5yQFXWepvr4+Kl2jHwgg4EGAxOABLwq7duvWTaZNmya6zlJxcbGsX78+Ct2iDwgg4EEgL5FIJDzsH/td8/LyZOPGjZFw0JVZN23aJNdff71UVVWJTlbzQgCB+AlQMcQv5u322L2k9f3335dRo0YxtNSuFB8gEG0BEkO045tx77RKSB5a2rx5c8bHYAcEEAi3AIkh3PELpPXuEt76XOnbb79d9NLW5ubmQM7FQRFAwD4BEoN9MbGmRfpc6ZUrV8qbb77J0JI1UaEhCAQvQGII3jjUZ3DvluaqpVCHkcYjkJEAVyVlxHXmxlG6KunM3rV9h6uW2nrwJwSiKkDFENXIBtAvrloKAJVDImChAInBwqDY3CT3qqUbbrjBuSFOn/HAWks2R4y2IZC5AM9jyNws9nvoVUv6jIdLL71UdK2lt956S374wx9Knz59Ym8DAAJREKBiiEIUDfXBXWvp2LFjzuNDWcbbUCA4LQI+C5AYfAaN2+F0rSV9fKi7jLf+ZGgpbt8C+hs1ARJD1CJqqD+6jPeKFSucex6uu+46ltMwFAdOi4AfAiQGPxQ5hiPQs2dPmT17tgwcOLB1pVaqB74cCIRPgPsYPMYsTvcxZEKl9zw899xzzkqtTExnIse2CJgXoGIwH4NItkDvedCHADExHcnw0qmIC5AYIh5gk91LNTHNYnwmI8K5EUhPgMSQnhNbeRDQiWldjG///v0sxufBkV0RyJUAiSFX0jE/z+l3TD/yyCNc1hrz7wTdt1eAyWePsWHyOXPAo0ePylNPPSXnnXeevPDCC6LzEbwQQMAeASoGe2IRm5boZa2LFi1yEkJhYaGsX7+e6iE20aejYRCgYvAYJSoGb4AHDx50EsMVV1zhzENQPXjzZG8E/BCgYvBDkWNkLeCut9SlSxfR6oFnTGdNyY4I+CZAxeCRkorBI2DS7gcOHJDq6mq59tprWa01yYVfEci1ABVDrsU5X7sC+ozpe++9t/WmOKqHdqn4AIFABUgMgfJy8EwF3JviFixYILfffrtMnTpVDh8+nOlh2B4BBDwIkBg84LFrcAJaPehNcZoU+vbt6wwxBXc2jowAAskCJIZkDX63SkBviisvL3ee9TBp0iQpKyujerAqQjQmqgIkhqhGNkL9cpfUeP/996keIhRXumKvAInB3tjQsiQBqockDH5FIGABEkPAwBzeXwGqB389ORoCqQS6pHqT96IhkDi6U5bdt0YOnNGdQTL+zr+TcSX9pSDvjA+tf8OtHgYPHiw69zBlyhTue7A+ajQwTAJUDGGKVoZtzes5UuavmisjC0QKxj8gazZulI3PPi6PfPsCqf3nKnli62/lRIbHtGlzt3pwr1zivgebokNbwixAYghz9NJoe16eyJ+Pf1Gu6n+x5Ov2eQXS//qr5So5Lu8eag51YtDuuNXD3LlznYX5uO8hjS8FmyDQiQCJoROgcH+ckJam9+Qd+ZJc2ff8U11JyIljv5ePRaSg4Fz5q3B3sLX1Wj1w13QrB78g4EmAxOCJz/adP5H39h+Q44MGS+HFfykiLXL0nW3yk0efk3evnCh3jRvweRVhezfSbF/yXdO6rPfNN98sjY2Nae7NZggg4Aow+exKRPFn4o/y0cFjIgfWyH1T13zewyvHy4yyRTJh0EDpeXYIZ57TiJPeNX3ZZZfJK6+84qzYum7dOpk8ebLk5zuDaWkcgU0QiLcAFUOE45/48D/kVwe6Ssncx2Tjs8tkxqACkS69pHDolZFNCm443erhoYcekiVLlsg3vvENqgcXh58IdCJAYugEKLwfJ+RE84dyyJ1fyDtPLup3ocihD6X5RCK83cqw5frgHx1WuuCCC5zqgWdNZwjI5rEUIDFENuyn5he+2F8uvUjnF7rJZV8eJAXHD8j+9z6JbK9Tdaxr165yyy23SGVlpTz77LPO8x7q6+tTbcp7CCAgIiSGqH4NWg7J/tr3pOCay6TnX2gn8yT/squkpOA9qd1/SFqi2u8O+qVPi9Pq4YYbbpDi4mK55557pLm5uYM9+AiBeAqQGCIZ9xZp2r1Lao8XSN9eBXK228f8vvLlksvk+M9ekVd+c0ziM6DkAoho9TBixAhZsWKFvPHGGzJy5Eipqan5/w34DQEEqBgi9x3483vy6j0zZP6anXJcjsuBNfNlzr/+5lSFcGo4SQ7Ivy2ZI1N/tFOOxjE7iEjPnj3l7rvvlhtvvFFKS0t5IFDk/iLQIS8CPPPZi54O0OTlycaNGz0ehd1NCuhw0vbt2+XVV1+VF198UcaPH8+lrSYDwrmNC5AYPIaAxOAR0KLdDxw44Dwp7vLLL3eeHqdXNPFCII4CzDHEMer0OaWA3hiny2pwaWtKHt6MkQAVg8dgUzF4BLR094MHD8pLL70kJ06ccCaqx44da2lLaRYC/gtQMfhvyhEjIKCXts6ePdu5tFUnp3X1Vl3emxcCcRAgMcQhyvQxKwH30taVK1fK/v37nedN6zMfWlrieBdIVoTsFFIBhpI8Bo6hJI+AIdqdyekQBYumehKgYvDEx85xEmByOk7RjndfqRg8xp+KwSNgSHd3J6d1WOnRRx8VJqdDGkianVKAiiElC28i0LGAOznNndMdO/GpDwIfVEtZXp5zM23e6CrZ+8lJOfnBLqkqGyR5eaNlxV7/F8WkYvAYNyoGj4AR2D35zmkeChSBgFrZhU/lSPU9MmzSr2Tqlgel56//IDc9+G0Z2C2Y/7cnMXj8EpAYPAJGaHd9jOhzzz0n5557rjz55JMyfPjwCPWOrhgX+MPrsmDgjbJMviNb9vxYJl7SNbAmBZNuAmsuB0bAXgH3oUDXX3+9lJSUcO+DvaEKZ8vO/7KUTr1aZNAQuapXcElBcUgM4fyK0GpLBdx7H3RZb+59sDRIYW3Wyf+W47//X5GabVL7nycC7QWJIVBeDh5XAV3Wu7y8XBYsWCD333+/88xpnhoX12+DH/3+VI689KS8+oWRMkp+Jw2H/Z9wTm4liSFZg98R8FlA733Qp8ZpouCpcT7jxuhwJ4/8TP6x5lr5h8oKmTq2SZ7dXidH9q6WB6oPyskAHEgMAaBySASSBXR4acyYMc5ifHV1ddKjRw9haY1koXj9Xl1dLTNmzEhr7a3P9q6QK/J6y40rRe5ZMUEu6dJTBo8ZIh8sWyErD46SByb2C2Q+gKuSPH4nuSrJI2AMd09eWmPp0qVSVFQUQ4X4dlmvXpsyZYrs2bNHtmzZIjfddJN1D4aiYojv95OeGxLQ4SWdd2B4yVAADJ9Wr17buXOnPPPMMzJp0iSZPHmyaLKw6UVisCkatCU2AgwvxSbUKTuan58v06ZNk4aGBufzwsJCefzxx61ZuZehpJRhS/9NhpLSt2LL9gV0eOmnP/2p6GNFly1bxvBS+1SR/ETnHbR6GDZsmDz99NPG459VYtB/DHkhgAACCAQjoMNMWlGYenXJ5sSJRCKb3SK5jyZJPCIZWmOd0vHm5cuXy9q1a51x6Ntuu826yUljOBE+sd7nMnPmTGdSun///kZ7yhyDUX5OjsCZAjo5uWbNGqmtrXXGnUeOHCm7du06c0PeiYSALsK4ZMkS5z6XUaNGyaFDh4yvs5XVUFIkouFTJ6gYfILkMCkF9HkPes9DWVmZzJs3z7mbWhMHr2gI1NTUyOLFi53O/OAHP7DmuR5UDNH4ftGLiAq4V6/o/0Xqy7arVyLKHni3Dh8+LPPnz5fS0lLRKmHbtm3WJAXtPBWDx68AFYNHQHbPSECHlHT+oampSWz6P8yMOhHzjXUuQZdHseUKpFThIDGkUsngPRJDBlhs6ouADi+99tprzuWNt956q3N5K8NLvtDm5CB6aaom9unTp1t7UQGJweNXgcTgEZDdsxbQSUtdUkMriMrKSpk1a5Z079496+OxIwKuAInBlcjyJ4khSzh2801AL2/V8Wr9v9CFCxdaufaOb53lQDkRIDF4ZCYxeARkd98EdIhCK4jevXvL97//feN3z/rWMQ6UcwGuSso5OSdEIBiBiRMnOle3fO1rX3MmN7WK0OEmXghkKkBiyFSM7RGwWEDnGCoqKpzF2Y4dO+Y8+2HDhg3WLM5mMR1NSxJgKCkJI5tfGUrKRo19ciWQfANVVVWV8Ttqc9VvzuNNgIrBmx97I2C1wNixY521/7WKKCkpcZ4cZtva/1YDxrRxJIaYBp5ux0cg+e7pCy+8kLun4xP6rHtKYsiajh0RCJdAnz59nJvh9LnTv2+2+VYAAAQ7SURBVPjFL0QX59OhJl4InC7AHMPpIhn+mTmGDMHY3AoB7p62IgzWNoKKwdrQ0DAEghPQ4SW9vPXjjz8WXU5DF+fTpZ+5vDU48zAdmcQQpmjRVgR8FtDLW/VRovrs4d27d8u4ceNEb5TjFW8BhpI8xp+hJI+A7G6VQPLd05owWJzPqvDkrDFUDDmj5kQI2C/g3j19zTXXMLxkf7gCayGJITBaDoxAOAV0eOmBBx5geCmc4fOl1QwleWRkKMkjILtbL8DwkvUh8r2BVAy+k3JABKIlwPBStOKZTm9IDOkosQ0CMRdgeCleXwCGkjzGm6Ekj4DsHkoBHV6aNGmSlJeXy7x587h6KZRRbL/RVAzt2/AJAgi0I+DeHMfaS+0AhfxtKgaPAaRi8AjI7qEXqK+vl5kzZzr9YGnv0IfT6QAVQzTiSC8QMCZQVFTUZmlvnhxnLBS+nZjE4BslB0IgvgLJS3u7T45jaY3wfh8YSvIYO4aSPAKyeyQF3CfH9e7d21mLiaU1whVmKoZwxYvWIhAKAX1y3LZt21pXbuW506EIW2sjqRhaKbL7hYohOzf2io9A8uT0008/LTonwctuASoGu+ND6xAIvYA7OT1t2jQpLi52nvugDwriZa8AicHe2NAyBCIjoJPTFRUVrQvz8VhRu0NLYrA7PrQOgUgJ6CT0pk2bnCRRWloqXNpqZ3hJDHbGhVYhEFkB99JWfWpcY2Oj89Q4vYqJlz0CJAZ7YkFLEIiVANWDveEmMdgbG1qGQOQFqB7sDDGJwc640CoEYiWQqnrgyiVzXwESgzl7zowAAkkCp1cPeuWS3gPBK/cCJIbcm3NGBBDoQCC5euC+hw6gAvyIO5894nLns0dAdkegAwG9amnKlCnOFhs3buSBQB1Y+fkRFYOfmhwLAQR8FdDqYefOnTJhwgQpLCxkaMlX3fYPRsXQvk1an1AxpMXERgh4FtDqQRMFr+AFSAwejUkMHgHZHQEErBNgKMm6kNAgBBBAwKwAicGsP2dHAAEErBMgMVgXEhqEAAIImBUgMZj15+wIIICAdQIkButCQoMQQAABswIkBrP+nB0BBBCwToDEYF1IaBACCCBgVoDEYNafsyOAAALWCZAYrAsJDUIAAQTMCpAYzPpzdgQQQMA6ARKDdSGhQQgggIBZARKDWX/OjgACCFgnQGKwLiQ0CAEEEDArQGIw68/ZEUAAAesESAzWhYQGIYAAAmYFSAxm/Tk7AgggYJ0AicG6kNAgBBBAwKwAicGsP2dHAAEErBMgMVgXEhqEAAIImBUgMZj15+wIIICAdQIkBo8hSSQSHo/A7ggggIBdAiQGu+JBaxBAAAHjAiQG4yGgAQgggIBdAiQGu+JBaxBAAAHjAiQG4yGgAQgggIBdAiQGu+JBaxBAAAHjAiQG4yGgAQgggIBdAiQGu+JBaxBAAAHjAv8Ho3mJlJUJ+y8AAAAASUVORK5CYII="></p>
<p>The region <em>R</em> is enclosed by the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</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 <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-axis. Find the area of <em>R</em>.</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> (limits in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>)</p>
<p>valid approach to find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-intercept <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = 0">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{6 - 2x}}{{\sqrt {16 + 6x - {x^2}} }} = 0">
<mfrac>
<mrow>
<mn>6</mn>
<mo>−</mo>
<mn>2</mn>
<mi>x</mi>
</mrow>
<mrow>
<msqrt>
<mn>16</mn>
<mo>+</mo>
<mn>6</mn>
<mi>x</mi>
<mo>−</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="6 - 2x = 0">
<mn>6</mn>
<mo>−</mo>
<mn>2</mn>
<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="x">
<mi>x</mi>
</math></span>-intercept is 3 <em><strong>(A1)</strong></em></p>
<p>valid approach using substitution or inspection <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u = 16 + 6x - {x^2}">
<mi>u</mi>
<mo>=</mo>
<mn>16</mn>
<mo>+</mo>
<mn>6</mn>
<mi>x</mi>
<mo>−</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^3 {\frac{{6 - 2x}}{{\sqrt u }}} {\text{d}}x">
<msubsup>
<mo>∫</mo>
<mn>0</mn>
<mn>3</mn>
</msubsup>
<mrow>
<mfrac>
<mrow>
<mn>6</mn>
<mo>−</mo>
<mn>2</mn>
<mi>x</mi>
</mrow>
<mrow>
<msqrt>
<mi>u</mi>
</msqrt>
</mrow>
</mfrac>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{d}}u = 6 - 2x">
<mrow>
<mtext>d</mtext>
</mrow>
<mi>u</mi>
<mo>=</mo>
<mn>6</mn>
<mo>−</mo>
<mn>2</mn>
<mi>x</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {\frac{1}{{\sqrt u }}} ">
<mo>∫</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mi>u</mi>
</msqrt>
</mrow>
</mfrac>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2{u^{\frac{1}{2}}}">
<mn>2</mn>
<mrow>
<msup>
<mi>u</mi>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
</math></span>,</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u = \sqrt {16 + 6x - {x^2}} ">
<mi>u</mi>
<mo>=</mo>
<msqrt>
<mn>16</mn>
<mo>+</mo>
<mn>6</mn>
<mi>x</mi>
<mo>−</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}u}}{{{\text{d}}x}} = \left( {6 - 2x} \right)\frac{1}{2}{\left( {16 + 6x - {x^2}} \right)^{ - \frac{1}{2}}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>u</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>6</mn>
<mo>−</mo>
<mn>2</mn>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>16</mn>
<mo>+</mo>
<mn>6</mn>
<mi>x</mi>
<mo>−</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int 2 \,{\text{d}}u">
<mo>∫</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>u</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2u">
<mn>2</mn>
<mi>u</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {f\left( x \right)} \,{\text{d}}x = 2\sqrt {16 + 6x - {x^2}} ">
<mo>∫</mo>
<mrow>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mn>2</mn>
<msqrt>
<mn>16</mn>
<mo>+</mo>
<mn>6</mn>
<mi>x</mi>
<mo>−</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</math></span> <em><strong>(A2)</strong></em></p>
<p>substituting <strong>both</strong> of <strong>their</strong> limits into <strong>their</strong> integrated function and subtracting <em><strong>(M1)</strong></em></p>
<p><em>eg </em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\sqrt {16 + 6\left( 3 \right) - {3^2}} - 2\sqrt {16 + 6{{\left( 0 \right)}^2} - {0^2}} ">
<mn>2</mn>
<msqrt>
<mn>16</mn>
<mo>+</mo>
<mn>6</mn>
<mrow>
<mo>(</mo>
<mn>3</mn>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mn>3</mn>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
<mo>−</mo>
<mn>2</mn>
<msqrt>
<mn>16</mn>
<mo>+</mo>
<mn>6</mn>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mn>0</mn>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\sqrt {16 + 18 - 9} - 2\sqrt {16} ">
<mn>2</mn>
<msqrt>
<mn>16</mn>
<mo>+</mo>
<mn>18</mn>
<mo>−</mo>
<mn>9</mn>
</msqrt>
<mo>−</mo>
<mn>2</mn>
<msqrt>
<mn>16</mn>
</msqrt>
</math></span></p>
<p><strong>Note:</strong> Award <em><strong>M0</strong></em> if they substitute into original or differentiated function. Do not accept only “– 0” as evidence of substituting lower limit.</p>
<p> </p>
<p>correct working <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\sqrt {25} - 2\sqrt {16} ">
<mn>2</mn>
<msqrt>
<mn>25</mn>
</msqrt>
<mo>−</mo>
<mn>2</mn>
<msqrt>
<mn>16</mn>
</msqrt>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="10 - 8">
<mn>10</mn>
<mo>−</mo>
<mn>8</mn>
</math></span></p>
<p>area = 2 <em><strong>A1 N2</strong></em></p>
<p> </p>
<p> </p>
<p><strong>METHOD 2</strong> (limits in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u">
<mi>u</mi>
</math></span>)</p>
<p>valid approach to find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-intercept <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = 0">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{6 - 2x}}{{\sqrt {16 + 6x - {x^2}} }} = 0">
<mfrac>
<mrow>
<mn>6</mn>
<mo>−</mo>
<mn>2</mn>
<mi>x</mi>
</mrow>
<mrow>
<msqrt>
<mn>16</mn>
<mo>+</mo>
<mn>6</mn>
<mi>x</mi>
<mo>−</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="6 - 2x = 0">
<mn>6</mn>
<mo>−</mo>
<mn>2</mn>
<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="x">
<mi>x</mi>
</math></span>-intercept is 3 <em><strong>(A1)</strong></em></p>
<p>valid approach using substitution or inspection <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u = 16 + 6x - {x^2}">
<mi>u</mi>
<mo>=</mo>
<mn>16</mn>
<mo>+</mo>
<mn>6</mn>
<mi>x</mi>
<mo>−</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^3 {\frac{{6 - 2x}}{{\sqrt u }}} {\text{d}}x">
<msubsup>
<mo>∫</mo>
<mn>0</mn>
<mn>3</mn>
</msubsup>
<mrow>
<mfrac>
<mrow>
<mn>6</mn>
<mo>−</mo>
<mn>2</mn>
<mi>x</mi>
</mrow>
<mrow>
<msqrt>
<mi>u</mi>
</msqrt>
</mrow>
</mfrac>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{d}}u = 6 - 2x">
<mrow>
<mtext>d</mtext>
</mrow>
<mi>u</mi>
<mo>=</mo>
<mn>6</mn>
<mo>−</mo>
<mn>2</mn>
<mi>x</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {\frac{1}{{\sqrt u }}} ">
<mo>∫</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mi>u</mi>
</msqrt>
</mrow>
</mfrac>
</mrow>
</math></span>, </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u = \sqrt {16 + 6x - {x^2}} ">
<mi>u</mi>
<mo>=</mo>
<msqrt>
<mn>16</mn>
<mo>+</mo>
<mn>6</mn>
<mi>x</mi>
<mo>−</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}u}}{{{\text{d}}x}} = \left( {6 - 2x} \right)\frac{1}{2}{\left( {16 + 6x - {x^2}} \right)^{ - \frac{1}{2}}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>u</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>6</mn>
<mo>−</mo>
<mn>2</mn>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>16</mn>
<mo>+</mo>
<mn>6</mn>
<mi>x</mi>
<mo>−</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int 2 \,{\text{d}}u">
<mo>∫</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>u</mi>
</math></span></p>
<p>correct integration <em><strong>(A2)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {\frac{1}{{\sqrt u }}} \,{\text{d}}u = 2{u^{\frac{1}{2}}}">
<mo>∫</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mi>u</mi>
</msqrt>
</mrow>
</mfrac>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>u</mi>
<mo>=</mo>
<mn>2</mn>
<mrow>
<msup>
<mi>u</mi>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int 2 \,{\text{d}}u = 2u">
<mo>∫</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>u</mi>
<mo>=</mo>
<mn>2</mn>
<mi>u</mi>
</math></span></p>
<p>both correct limits for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u">
<mi>u</mi>
</math></span> <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u">
<mi>u</mi>
</math></span> = 16 and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u">
<mi>u</mi>
</math></span> = 25, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_{16}^{25} {\frac{1}{{\sqrt u }}{\text{d}}u} ">
<msubsup>
<mo>∫</mo>
<mrow>
<mn>16</mn>
</mrow>
<mrow>
<mn>25</mn>
</mrow>
</msubsup>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mi>u</mi>
</msqrt>
</mrow>
</mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>u</mi>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left[ {2{u^{\frac{1}{2}}}} \right]_{16}^{25}">
<msubsup>
<mrow>
<mo>[</mo>
<mrow>
<mn>2</mn>
<mrow>
<msup>
<mi>u</mi>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
</mrow>
<mo>]</mo>
</mrow>
<mrow>
<mn>16</mn>
</mrow>
<mrow>
<mn>25</mn>
</mrow>
</msubsup>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u">
<mi>u</mi>
</math></span> = 4 and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u">
<mi>u</mi>
</math></span> = 5, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_4^5 2 \,{\text{d}}u">
<msubsup>
<mo>∫</mo>
<mn>4</mn>
<mn>5</mn>
</msubsup>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>u</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left[ {2u} \right]_4^5">
<msubsup>
<mrow>
<mo>[</mo>
<mrow>
<mn>2</mn>
<mi>u</mi>
</mrow>
<mo>]</mo>
</mrow>
<mn>4</mn>
<mn>5</mn>
</msubsup>
</math></span></p>
<p>substituting <strong>both</strong> of <strong>their</strong> limits for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u">
<mi>u</mi>
</math></span> (do not accept 0 and 3) into <strong>their</strong> integrated function and subtracting <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\sqrt {25} - 2\sqrt {16} ">
<mn>2</mn>
<msqrt>
<mn>25</mn>
</msqrt>
<mo>−</mo>
<mn>2</mn>
<msqrt>
<mn>16</mn>
</msqrt>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="10 - 8">
<mn>10</mn>
<mo>−</mo>
<mn>8</mn>
</math></span></p>
<p><strong>Note:</strong> Award <em><strong>M0</strong> </em>if they substitute into original or differentiated function, or if they have not attempted to find limits for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u">
<mi>u</mi>
</math></span>.</p>
<p> </p>
<p>area = 2 <em><strong>A1 N2</strong></em></p>
<p> </p>
<p><strong><em>[8 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p>The derivative of a function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is given by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'(x) = 2{{\text{e}}^{ - 3x}}">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>3</mn>
<mi>x</mi>
</mrow>
</msup>
</mrow>
</math></span>. The graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> passes through <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\frac{1}{3}{\text{,}}\,\,5} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mn>5</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<p>Find <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>.</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>recognizing to integrate <em><strong>(M1)</strong></em></p>
<p><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {f'} ">
<mo>∫</mo>
<mrow>
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {2{{\text{e}}^{ - 3x}}{\text{d}}x} ">
<mo>∫</mo>
<mrow>
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>3</mn>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{d}}u = - 3">
<mrow>
<mtext>d</mtext>
</mrow>
<mi>u</mi>
<mo>=</mo>
<mo>−</mo>
<mn>3</mn>
</math></span></p>
<p>correct integral (do not penalize for missing +<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C">
<mi>C</mi>
</math></span>) <em><strong>(A2)</strong><br></em></p>
<p><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \frac{2}{3}{{\text{e}}^{ - 3x}} + C">
<mo>−</mo>
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>3</mn>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mi>C</mi>
</math></span></p>
<p>substituting <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\frac{1}{3}{\text{,}}\,\,5} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mn>5</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> (in any order) into <strong>their</strong> integrated expression (must have +<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C">
<mi>C</mi>
</math></span>) <em><strong>M1</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \frac{2}{3}{{\text{e}}^{ - 3\left( {1/3} \right)}} + C = 5">
<mo>−</mo>
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>3</mn>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mrow>
<mo>/</mo>
</mrow>
<mn>3</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mi>C</mi>
<mo>=</mo>
<mn>5</mn>
</math></span></p>
<p><strong>Note:</strong> Award <em><strong>M0</strong> </em>if they substitute into original or differentiated function.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = - \frac{2}{3}{{\text{e}}^{ - 3x}} + 5 + \frac{2}{3}{{\text{e}}^{ - 1}}">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>3</mn>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mn>5</mn>
<mo>+</mo>
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span> (or <em>any</em> equivalent form, <em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \frac{2}{3}{{\text{e}}^{ - 3x}} + 5 - \frac{2}{{ - 3{\text{e}}}}">
<mo>−</mo>
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>3</mn>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mn>5</mn>
<mo>−</mo>
<mfrac>
<mn>2</mn>
<mrow>
<mo>−</mo>
<mn>3</mn>
<mrow>
<mtext>e</mtext>
</mrow>
</mrow>
</mfrac>
</math></span>) <em><strong>A1 N4</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br>