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<h2>HL Paper 2</h2><div class="specification">
<p>A wind turbine is designed so that the rotation of the blades generates electricity. The turbine is built on horizontal ground and is made up of a vertical tower and three blades.</p>
<p>The point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> is on the base of the tower directly below point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> at the top of the tower. The height of the tower, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>AB</mtext></math>, is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>90</mn><mo> </mo><mtext>m</mtext></math>. The blades of the turbine are centred at <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> and are each of length <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>40</mn><mo> </mo><mtext>m</mtext></math>. This is shown in the following diagram.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" 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"></p>
<p>The end of one of the blades of the turbine is represented by point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>C</mtext></math> on the diagram. Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math> be the height of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>C</mtext></math> above the ground, measured in metres, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math> varies as the blade rotates.</p>
</div>
<div class="specification">
<p>Find the</p>
</div>
<div class="specification">
<p>The blades of the turbine complete <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn></math> rotations per minute under normal conditions, moving at a constant rate.</p>
</div>
<div class="specification">
<p>The height, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math>, of point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>C</mtext></math> can be modelled by the following function. Time, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>, is measured from the instant when the blade <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>[BC]</mtext></math> first passes <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>[AB]</mtext></math> and is measured in seconds.</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mn>90</mn><mo>-</mo><mn>40</mn><mo> </mo><mi>cos</mi><mfenced><mrow><mn>72</mn><mi>t</mi><mo>°</mo></mrow></mfenced><mo>,</mo><mo> </mo><mi>t</mi><mo>≥</mo><mn>0</mn></math></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>maximum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</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>minimum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the time, in seconds, it takes for the blade <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>[BC]</mtext></math> to make one complete rotation under these conditions.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the angle, in degrees, that the blade <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>[BC]</mtext></math> turns through in one second.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the amplitude of the function.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the period of the function.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>(</mo><mi>t</mi><mo>)</mo></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>ā¤</mo><mi>t</mi><mo>ā¤</mo><mn>5</mn></math>, clearly labelling the coordinates of the maximum and minimum points.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the height of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>C</mtext></math> above the ground 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">e.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the time, in seconds, that point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>C</mtext></math> is above a height of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn><mo> </mo><mtext>m</mtext></math>, during each complete rotation.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The wind speed increases and the blades rotate faster, but still at a constant rate.</p>
<p>Given that point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>C</mtext></math> is now higher than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>110</mn><mo> </mo><mtext>m</mtext></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> second during each complete rotation, find the time for one complete rotation.</p>
<div class="marks">[5]</div>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p>The cross-sectional view of a tunnel is shown on the axes below. The line <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><mtext>AB</mtext><mo>]</mo></math> represents a vertical wall located at the left side of the tunnel. The height, in metres, of the tunnel above the horizontal ground is modelled by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mo> </mo><mn>0</mn><mo>.</mo><mn>8</mn><msup><mi>x</mi><mn>2</mn></msup><mo>,</mo><mo> </mo><mn>2</mn><mo>ā¤</mo><mi>x</mi><mo>ā¤</mo><mn>8</mn></math>, relative to an origin <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math>.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">Point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> has coordinates <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>2</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math>, point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> has coordinates <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>2</mn><mo>,</mo><mo> </mo><mn>2</mn><mo>.</mo><mn>4</mn><mo>)</mo></math>, and point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>C</mtext></math> has coordinates <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>8</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math>.</p>
</div>
<div class="specification">
<p>Find the height of the tunnel when</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the maximum height of the tunnel.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>4</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>6</mn></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the trapezoidal rule, with three intervals, to estimate the cross-sectional area of the tunnel.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the integral which can be used to find the cross-sectional area of the tunnel.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the cross-sectional area of the tunnel.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>A student investigating the relationship between chemical reactions and temperature finds the Arrhenius equation on the internet.</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mi>A</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mfrac><mi>c</mi><mi>T</mi></mfrac></mrow></msup></math></p>
<p>This equation links a variable <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> with the temperature <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math> are positive constants and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>></mo><mn>0</mn></math>.</p>
</div>
<div class="specification">
<p>The Arrhenius equation predicts that the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>k</mi></math> against <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mi>T</mi></mfrac></math> is a straight line.</p>
</div>
<div class="specification">
<p>Write down</p>
</div>
<div class="specification">
<p>The following data are found for a particular reaction, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math> is measured in Kelvin and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> is measured in <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>cm</mtext><mn>3</mn></msup><mo> </mo><msup><mtext>mol</mtext><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo> </mo><msup><mtext>s</mtext><mrow><mo>−</mo><mn>1</mn></mrow></msup></math>:</p>
<p style="text-align: center;"><img 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"></p>
</div>
<div class="specification">
<p>Find an estimate of</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>k</mi></mrow><mrow><mo>d</mo><mi>T</mi></mrow></mfrac></math> is always positive.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>T</mi><mo>ā</mo><mo>ā</mo></mrow></munder><mi>k</mi><mo>=</mo><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>T</mi><mo>ā</mo><mn>0</mn></mrow></munder><mi>k</mi><mo>=</mo><mn>0</mn></math>, sketch the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> against <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>(i) the gradient of this line in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math>;</p>
<p>(ii) the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-intercept of this line in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of the regression line for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo>ā</mo><mi>k</mi></math> on <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mi>T</mi></mfrac></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math>.</p>
<p>It is not required to state units for this value.</p>
<div class="marks">[1]</div>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math>.</p>
<p>It is not required to state units for this value.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the expression <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = {\text{tan}}\left( {x + \frac{\pi }{4}} \right){\text{cot}}\left( {\frac{\pi }{4} - x} \right)">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mtext>tan</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mfrac>
<mi>Ļ<!-- Ļ --></mi>
<mn>4</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>cot</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>Ļ<!-- Ļ --></mi>
<mn>4</mn>
</mfrac>
<mo>ā<!-- ā --></mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>The expression <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right)">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> can be written as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( t \right)">
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = {\text{tan}}\,x">
<mi>t</mi>
<mo>=</mo>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</math></span>.</p>
</div>
<div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha ">
<mi>Ī±<!-- Ī± --></mi>
</math></span>, <em>Ī²</em> be the roots of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( t \right) = k">
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>k</mi>
</math></span>, where 0 < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span> < 1.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( x \right)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \frac{{5\pi }}{8} \leqslant x \leqslant \frac{\pi }{8}">
<mo>ā</mo>
<mfrac>
<mrow>
<mn>5</mn>
<mi>Ļ</mi>
</mrow>
<mn>8</mn>
</mfrac>
<mo>ā©½</mo>
<mi>x</mi>
<mo>ā©½</mo>
<mfrac>
<mi>Ļ</mi>
<mn>8</mn>
</mfrac>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>With reference to your graph, explain why <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is a function on the given domain.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain why <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> has no inverse on the given domain.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain why <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is not a function for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \frac{{3\pi }}{4} \leqslant x \leqslant \frac{\pi }{4}">
<mo>ā</mo>
<mfrac>
<mrow>
<mn>3</mn>
<mi>Ļ</mi>
</mrow>
<mn>4</mn>
</mfrac>
<mo>ā©½</mo>
<mi>x</mi>
<mo>ā©½</mo>
<mfrac>
<mi>Ļ</mi>
<mn>4</mn>
</mfrac>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.iv.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( t \right) = {\left( {\frac{{1 + t}}{{1 - t}}} \right)^2}">
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>t</mi>
</mrow>
<mrow>
<mn>1</mn>
<mo>ā</mo>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = g\left( t \right)">
<mi>y</mi>
<mo>=</mo>
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> for <em>t</em> ā¤ 0. Give the coordinates of any intercepts and the equations of any asymptotes.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha ">
<mi>Ī±</mi>
</math></span> and <em>Ī²</em> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha ">
<mi>Ī±</mi>
</math></span> + <em>Ī²</em> < ā2.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>Charlotte decides to model the shape of a cupcake to calculate its volume.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p>From rotating a photograph of her cupcake she estimates that its cross-section passes through the points <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>3</mn><mo>.</mo><mn>5</mn><mo>)</mo><mo>,</mo><mo> </mo><mo>(</mo><mn>4</mn><mo>,</mo><mo> </mo><mn>6</mn><mo>)</mo><mo>,</mo><mo> </mo><mo>(</mo><mn>6</mn><mo>.</mo><mn>5</mn><mo>,</mo><mo> </mo><mn>4</mn><mo>)</mo><mo>,</mo><mo> </mo><mo>(</mo><mn>7</mn><mo>,</mo><mo> </mo><mn>3</mn><mo>)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>7</mn><mo>.</mo><mn>5</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math>, where all units are in centimetres. The cross-section is symmetrical in the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis, as shown below:</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p>She models the section from <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>3</mn><mo>.</mo><mn>5</mn><mo>)</mo></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>4</mn><mo>,</mo><mo> </mo><mn>6</mn><mo>)</mo></math> as a straight line.</p>
</div>
<div class="specification">
<p>Charlotte models the section of the cupcake that passes through the points <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>4</mn><mo>,</mo><mo> </mo><mn>6</mn><mo>)</mo><mo>,</mo><mo> </mo><mo>(</mo><mn>6</mn><mo>.</mo><mn>5</mn><mo>,</mo><mo> </mo><mn>4</mn><mo>)</mo><mo>,</mo><mo> </mo><mo>(</mo><mn>7</mn><mo>,</mo><mo> </mo><mn>3</mn><mo>)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>7</mn><mo>.</mo><mn>5</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math> with a quadratic curve.</p>
</div>
<div class="specification">
<p>Charlotte thinks that a quadratic with a maximum point at <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>4</mn><mo>,</mo><mo> </mo><mn>6</mn><mo>)</mo></math> and that passes through the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>7</mn><mo>.</mo><mn>5</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math> would be a better fit.</p>
</div>
<div class="specification">
<p>Believing this to be a better model for her cupcake, Charlotte finds the volume of revolution about the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis to estimate the volume of the cupcake.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of the line passing through these two points.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of the least squares regression quadratic curve for these four points.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By considering the gradient of this curve when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>4</mn></math>, explain why it may not be a good model.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of the new model.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down an expression for her estimate of the volume as a sum of two integrals.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of Charlotteās estimate.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>An environmental scientist is asked by a river authority to model the effect of a leak from a power plant on the mercury levels in a local river. The variable <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> measures the concentration of mercury in micrograms per litre.</p>
<p>The situation is modelled using the second order differential equation</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mtext>d</mtext><mn>2</mn></msup><mi>x</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mn>3</mn><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>+</mo><mn>2</mn><mi>x</mi><mo>=</mo><mn>0</mn></math></p>
<p>where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>≥</mo><mn>0</mn></math> is the time measured in days since the leak started. It is known that when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>=</mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mn>1</mn></math>.</p>
</div>
<div class="specification">
<p>If the mercury levels are greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>1</mn></math> micrograms per litre, fishing in the river is considered unsafe and is stopped.</p>
</div>
<div class="specification">
<p>The river authority decides to stop people from fishing in the river for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo>%</mo></math> longer than the time found from the model.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the system of coupled first order equations:</p>
<p style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>y</mi></math></p>
<p style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>3</mn><mi>y</mi></math></p>
<p style="text-align:left;">can be written as the given second order differential equation.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the eigenvalues of the system of coupled first order equations given in part (a).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the exact solution of the second order differential equation.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> against <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>, labelling the maximum point of the graph with its coordinates.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the model to calculate the total amount of time when fishing should be stopped.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down one reason, with reference to the context, to support this decision.</p>
<div class="marks">[1]</div>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p>The voltage <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v">
<mi>v</mi>
</math></span> in a circuit is given by the equation</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v\left( t \right) = 3\,{\text{sin}}\left( {100\pi t} \right)">
<mi>v</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>100</mn>
<mi>Ļ<!-- Ļ --></mi>
<mi>t</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t \geqslant 0">
<mi>t</mi>
<mo>ā©¾<!-- ā©¾ --></mo>
<mn>0</mn>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> is measured in seconds.</p>
</div>
<div class="specification">
<p>The current <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="i">
<mi>i</mi>
</math></span> in this circuit is given by the equation</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="i\left( t \right) = 2\,{\text{sin}}\left( {100\pi \left( {t + 0.003} \right)} \right)">
<mi>i</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>100</mn>
<mi>Ļ<!-- Ļ --></mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>+</mo>
<mn>0.003</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>The power <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span> in this circuit is given by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\left( t \right) = v\left( t \right) \times i\left( t \right)">
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>v</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>Ć<!-- Ć --></mo>
<mi>i</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>The average power <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{p_{av}}">
<mrow>
<msub>
<mi>p</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
</mrow>
</msub>
</mrow>
</math></span> in this circuit from <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 0">
<mi>t</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = T">
<mi>t</mi>
<mo>=</mo>
<mi>T</mi>
</math></span> is given by the equation</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{p_{av}}\left( T \right) = \frac{1}{T}\int_0^T {p\left( t \right){\text{d}}t} ">
<mrow>
<msub>
<mi>p</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mi>T</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mi>T</mi>
</mfrac>
<msubsup>
<mo>ā«<!-- ā« --></mo>
<mn>0</mn>
<mi>T</mi>
</msubsup>
<mrow>
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="T > 0">
<mi>T</mi>
<mo>></mo>
<mn>0</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the maximum and minimum value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v">
<mi>v</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down two transformations that will transform the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = v\left( t \right)">
<mi>y</mi>
<mo>=</mo>
<mi>v</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> onto the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = i\left( t \right)">
<mi>y</mi>
<mo>=</mo>
<mi>i</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = p\left( t \right)">
<mi>y</mi>
<mo>=</mo>
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> for 0 ā¤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> ā¤ 0.02 , showing clearly the coordinates of the first maximum and the first minimum.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the total time in the interval 0 ā¤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> ā¤ 0.02 for which <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\left( t \right)">
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> ā„ 3.</p>
<p> </p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{p_{av}}">
<mrow>
<msub>
<mi>p</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
</mrow>
</msub>
</mrow>
</math></span>(0.007).</p>
<p> </p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>With reference to your graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = p\left( t \right)">
<mi>y</mi>
<mo>=</mo>
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> explain why <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{p_{av}}\left( T \right)">
<mrow>
<msub>
<mi>p</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mi>T</mi>
<mo>)</mo>
</mrow>
</math></span> > 0 for all <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="T">
<mi>T</mi>
</math></span> > 0.</p>
<p> </p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\left( t \right)">
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> can be written as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\left( t \right) = a\,{\text{sin}}\left( {b\left( {t - c} \right)} \right) + d">
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>a</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>b</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>ā</mo>
<mi>c</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>d</mi>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c">
<mi>c</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d">
<mi>d</mi>
</math></span> > 0, use your graph to find the values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c">
<mi>c</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d">
<mi>d</mi>
</math></span>.</p>
<p> </p>
<div class="marks">[6]</div>
<div class="question_part_label">g.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = - 1 + \ln \left( {\sqrt {{x^2} - 1} } \right)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>ā<!-- ā --></mo>
<mn>1</mn>
<mo>+</mo>
<mi>ln</mi>
<mo>ā”<!-- ā” --></mo>
<mrow>
<mo>(</mo>
<mrow>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>ā<!-- ā --></mo>
<mn>1</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
</div>
<div class="specification">
<p>The function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = - 1 + \ln \left( {\sqrt {{x^2} - 1} } \right),{\text{ }}x \in D">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>ā<!-- ā --></mo>
<mn>1</mn>
<mo>+</mo>
<mi>ln</mi>
<mo>ā”<!-- ā” --></mo>
<mrow>
<mo>(</mo>
<mrow>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>ā<!-- ā --></mo>
<mn>1</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>ā<!-- ā --></mo>
<mi>D</mi>
</math></span></p>
</div>
<div class="specification">
<p>The function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g">
<mi>g</mi>
</math></span> is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g(x) = - 1 + \ln \left( {\sqrt {{x^2} - 1} } \right),{\text{ }}x \in \left] {1,{\text{ }}\infty } \right[">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>ā<!-- ā --></mo>
<mn>1</mn>
<mo>+</mo>
<mi>ln</mi>
<mo>ā”<!-- ā” --></mo>
<mrow>
<mo>(</mo>
<mrow>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>ā<!-- ā --></mo>
<mn>1</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>ā<!-- ā --></mo>
<mrow>
<mo>]</mo>
<mrow>
<mn>1</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi mathvariant="normal">ā<!-- ā --></mi>
</mrow>
<mo>[</mo>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the largest possible domain <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="D">
<mi>D</mi>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> to be a function.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f(x)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span> showing clearly the equations of asymptotes and the coordinates of any intercepts with the axes.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain why <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is an even function.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain why the inverse function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f^{ - 1}}">
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mo>ā</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span> does not exist.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the inverse function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{g^{ - 1}}">
<mrow>
<msup>
<mi>g</mi>
<mrow>
<mo>ā</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span> and state its domain.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'(x)">
<msup>
<mi>g</mi>
<mo>ā²</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, show that there are no solutions to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'(x) = 0">
<msup>
<mi>g</mi>
<mo>ā²</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</math></span>;</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, show that there are no solutions to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="({g^{ - 1}})'(x) = 0">
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>g</mi>
<mrow>
<mo>ā</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mo>ā²</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>At an archery tournament, a particular competition sees a ball launched into the air while an archer attempts to hit it with an arrow.</p>
<p>The path of the ball is modelled by the equation</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mn>5</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>t</mi><mfenced><mtable><mtr><mtd><msub><mi>u</mi><mi>x</mi></msub></mtd></mtr><mtr><mtd><msub><mi>u</mi><mi>y</mi></msub><mo>-</mo><mn>5</mn><mi>t</mi></mtd></mtr></mtable></mfenced></math></p>
<p>where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> is the horizontal displacement from the archer and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> is the vertical displacement from the ground, both measured in metres, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> is the time, in seconds, since the ball was launched.</p>
<ul>
<li><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>x</mi></msub></math> is the horizontal component of the initial velocity</li>
<li><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>y</mi></msub></math> is the vertical component of the initial velocity.</li>
</ul>
<p>In this question both the ball and the arrow are modelled as single points. The ball is launched with an initial velocity such that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>x</mi></msub><mo>=</mo><mn>8</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>y</mi></msub><mo>=</mo><mn>10</mn></math>.</p>
</div>
<div class="specification">
<p>An archer releases an arrow from the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>2</mn><mo>)</mo></math>. The arrow is modelled as travelling in a straight line, in the same plane as the ball, with speed <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>60</mn><mo> </mo><msup><mtext>m s</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> and an angle of elevation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo>°</mo></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the initial speed of the ball.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the angle of elevation of the ball as it is launched.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the maximum height reached by the ball.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Assuming that the ground is horizontal and the ball is not hit by the arrow, find the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> coordinate of the point where the ball lands.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>For the path of the ball, find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the two positions where the path of the arrow intersects the path of the ball.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the time when the arrow should be released to hit the ball before the ball reaches its maximum height.</p>
<div class="marks">[4]</div>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graphs <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {\text{si}}{{\text{n}}^3}\,x + {\text{ln}}\,x">
<mi>y</mi>
<mo>=</mo>
<mrow>
<mtext>si</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>3</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 1 + {\text{cos}}\,x">
<mi>y</mi>
<mo>=</mo>
<mn>1</mn>
<mo>+</mo>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</math></span> on the following axes for 0 < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> ā¤ 9.</p>
<p style="text-align: center;"><img 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"></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence solve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{si}}{{\text{n}}^3}\,x + {\text{ln}}\,x - {\text{cos}}\,x - 1 < 0">
<mrow>
<mtext>si</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>3</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>ā</mo>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>ā</mo>
<mn>1</mn>
<mo><</mo>
<mn>0</mn>
</math></span> in the range 0 < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> ā¤ 9.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = \frac{{\sqrt x }}{{\sin x}},{\text{ }}0 < x < \pi ">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mrow>
<msqrt>
<mi>x</mi>
</msqrt>
</mrow>
<mrow>
<mi>sin</mi>
<mo>ā”<!-- ā” --></mo>
<mi>x</mi>
</mrow>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>0</mn>
<mo><</mo>
<mi>x</mi>
<mo><</mo>
<mi>Ļ<!-- Ļ --></mi>
</math></span>.</p>
</div>
<div class="specification">
<p>Consider the region bounded by the curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f(x)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span>, the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis and the lines <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{\pi }{6},{\text{ }}x = \frac{\pi }{3}">
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mi>Ļ<!-- Ļ --></mi>
<mn>6</mn>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mi>Ļ<!-- Ļ --></mi>
<mn>3</mn>
</mfrac>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-coordinate of the minimum point on the curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f(x)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span> satisfies the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\tan x = 2x">
<mi>tan</mi>
<mo>ā”</mo>
<mi>x</mi>
<mo>=</mo>
<mn>2</mn>
<mi>x</mi>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> for which <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span> is a decreasing function.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f(x)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span> showing clearly the minimum point and any asymptotic behaviour.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the coordinates of the point on the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> where the normal to the graph is parallel to the line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = - x">
<mi>y</mi>
<mo>=</mo>
<mo>ā</mo>
<mi>x</mi>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>This region is now rotated through <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\pi ">
<mn>2</mn>
<mi>Ļ</mi>
</math></span> radians about the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis. Find the volume of revolution.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Beth goes for a run. She uses a fitness app to record her distance, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s">
<mi>s</mi>
</math></span>ākm, and time, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span>āminutes. A graph of her distance against time is shown.</p>
<p><img src="data:image/png;base64,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"></p>
<p>Beth runs at a constant speed of 2.3āms<sup>ā1</sup> for the first 8 minutes.</p>
</div>
<div class="specification">
<p>Between 8 and 20 minutes, her distance can be modeled by a cubic function, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s = a{t^3} + b{t^2} + ct + d">
<mi>s</mi>
<mo>=</mo>
<mi>a</mi>
<mrow>
<msup>
<mi>t</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>b</mi>
<mrow>
<msup>
<mi>t</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>c</mi>
<mi>t</mi>
<mo>+</mo>
<mi>d</mi>
</math></span>. She reads the following data from her app.</p>
<p style="text-align: center;"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAASQAAAA1CAYAAADoBJZmAAAPjUlEQVR4Ae2d3ctVxRfHtz9+t5maV5EhvoCikaRWVAYKZmh3GSZ0IRSYL3eFliFdRJmRXoj4cuGdYmEghC+9QBc+EpomBnqnIQpdaS/6Bzw/PvPje1pn3HvOnHNmP2c/z5mBc2bPzJo1a62ZWfOy18yeNDo6OlpklyWQJZAl0AAJ/KcBNGQSsgSyBLIEnASyQsoNIUsgS6AxEsgKqTFVkQnJEsgSyAopt4EsgSyBxkggK6TGVEUmJEsgS+C/nUQwadKkTiA5PUsgSyBLoCsJVL3c76iQKKUqc1cUZOBkEmCQGJY6GRZeh4VPOkFokpOXbMnUREaUJZAl0K8EskLqV4I5f5ZAlkAyCWSFlEyUGVGWQJZAvxIYmEL67bffijfffNOtJ7/44ot++RiX+eGb9TRyOH/+/LjkIROdJZBSAgNRSCijRYsWFc8880xx9+7dYtu2bX3z9Oqrrxb8xpODb/ifPXt2sWzZslqV0pkzZ4o5c+ZUioc6QX4oyGnTphWbN28u7t27Vwnf5IQQr7///nvBQLB06VLHqwYE4sebC/FpeWGwoz6p148++sgmNe+Zw7Uh9/+XbCGI9jTgd+/e3R7phdatWze6atUqL7a/IPhS47QUXb16lUPIoyMjIzY62fOSJUtGkUuM66ZOoBu5kKcq382bN0enTp06umnTJlc8PBKOpSeG5l5hqmguwxfDK3zxUz3iU0adbaeMVj8uNZ/CT1+EX3zquQkuxCuvj4MulNnPePz48ahOC85Dhw752RsdjuWtVyZoMLGyjoWjAaJU8EP4d+zY4RqtpR3lRDmDbsSpeQUfdWmdFLaNG+vn1HxCv5QRirpJLsRrMoU0e/Zs14ApTL+yUUcjkkYoCUodnrx37951ozXPaHfSyuKUV53Hlid6qJTTp0+PKgwMuOTATznAyQlWceqw4ku+4OGFGY6lV2n4KF/hBAZYv5FUycXi0TM4unXwUpUP2qzswI3MgPc7b7fl9gtfRXMIb4jXsnzw3ks5Zbh6jeul/BCfDCTgVBvula468oV4TbaHdOPGjWLVqlXuh9Eev++++y56jcrG7o4dOxz8zp07i/fff7+1v8L6tyxOyA8cOFAsWbJEQedfvHjR+SdOnCju3LlTQN/p06eL77//vjhy5EgL9ttvv20960F5Ff7000+LDz74wAVHRkYcbzJMZH3O/s8777zj4o8ePVqsX7++tR/EOv/dd98t9u3b59LJf/ny5eLBgwdCP3D/5s2bxaxZs9romDx5sgvfvn27LX4iBtR2JxJvX375pWPn7bffHldsJVNIcE1nX7FiRVAAP//8s0t/6aWXHoJ79NFHXRzKiA7y2GOPFW+88Ubx119/Fa+99tpDcXYjEljrFGbzcuPGjS5p9erVxdSpU4uffvrJgj70rLwPJZRE7N+/3ylDWwbKkXjctWvXnD9v3jznw/e6devcs/2TPCQfmzYWzzNnzhyLYhpXxuHDh4s///yz2L17d+No64egr776yrVLBl9eZuhlRdPfaCdTSFIOL7zwQj9ydHn90ZpIjdgW+R9//GGDpc9+R3v22WdL4XqN/Prrr53StPlfeeWVglkHbu3atU4JoqRoDLy5orFIAdl8+XlsJcCbxQ8//LBghv3000+PbeE1l8YgTht88sknC2b8zOiZ5W/fvr1g1t5Ul0wh/fLLL47HYexoVDIjkH6fffaZW5YhEJQrSzSWpMDNnTvXKaSmNYgrV640jaRa6WEAXb58uVNG1M1EdCggeNOMHzMTVginTp1qLLvJFNK5c+ecPU1jOa2RMKb72jezvopEKTEKY3PE7Ik9Jkbnpjga6d9//11KzsKFC0vjx3Mks1SWzeqw45mXbmlnhaDVTLd5xwI+mUKCyZDh3Vgw00sZZTO6MoPAqqUoS7HQ7AJjQ02RGam2bt3qyLxw4UIv5NaSByXJ/p/l++zZs66s5557rpYyB4l0y5YtzjDSGuRiMDiRrOUxtuWFju9YySxevNiPbkw4mULiTYUcyonlS5lTxy6r/H/++cdlsRpccffv32+hK4ujM1ka1Llu3brVyscDMEpTAjMEKo94yqbB4lQOz4888oiLU0flzR+w7733XsE+EpujOHDwbDcPP/7449aopE3r559/3sHrT/KQfBSfyhcvKsfilZLkTSaO2dvBgwfdRq+m+xa+6c8hXtm/Ywn9ySeftNigzqjD8eZCfPJWFz5tu6TN4hr95q2TnUHIZsDmtXZE2LVUuSp7G5sf2yActjqUz484wmVxskMCTtbF1u5HRpikCZ+skimHsmWPRDx2SoKzdjjYIxEPbux05LD1UHn41vaDsmWjVJZXOKrkonTrgyfWYWMj2sQTcZYvcFlbLWRheYgtqw641Lz6spBM8KmDQbnUfMKHb/9G+/ft3wbBb4jXSRAUGhmY6XQACWUvTQMn+y52ylwKOESRjGTYK8XIuo46aaqoh4XXYeGTdhbiNdmSrZsGzYZi2fq2GxwTDfbkyZOl9kkTjc/MT5ZASAIDUUjsWbC+ZSPR388JETsR0+Cf/SY2lbGJyS5LYJglELVkG2YBZd6zBLIE0kugamsiX/KfXta1YwytwWsvfIwLGBZeh4VPmg+8VrmBLNmqiMnxWQJZAsMtgayQhrv+M/dZAo2SQFZIjaqOTEyWwHBLYKAKiSMVutsYC2Jdk1DH3di8yWLtymHDMmvl4W4GmfthlABveOvoa/3IcmAKCUPAt956yx29YMedM2Uc6+AMTh1uLC/Ur4P+fnGi/GPPGnJ0RJfCo8SrLv0HDgUPTAjODjzA0QlSHy6mc3EsBNyih8EuZvCRbMiHjOyxHyt34pEFcOAmn++gA9mJBuSTklfxqYE8RIulzcpFtE2fPt2CtJ5j+BSwZKdw334n0/GQmXenvKH0quMJHGvgV6fr5kL9OunoFXc3dcJRAeRJnph8OprDMQNd9csRBPLa4zaC4ziN4Dh6Qb3a+tOxHB1VARY8wMXc1x1DM3LUsSAdEwI3x0QoR/SVydu/qle8Co/y6Bpj+BYP0EbYOtoWP2CggWfoCNFA/m751DEXlUH+kDxj+1Usn/DdTbuyMgrx2vFQVCizLaSbZ53b8isTHLGC66Y8H5ZzWnXw5ZdTVziWdhooHRU/lmfVjd+4KdMqmip8fjwKgUZuHZ0TfH68hdFzLK/QZhUm+UWLOq9wWr9scEKB8JMTvVZJIR9os2VKudl2jSIGTgpZOH2/bj5j+lUsn720K8tviNekSzbt02hKWLVE0In3mFv6NEUGJ8s8puXCb6fHwJFWFudPI3WiPmY67+cdT2HuYUImZTdwVvHB0pkB2+bRkmPDhg1V2driuT1BjpsLdTWx4rhBgGtbfv31V0X17XN/O3dOdeO4rYETA3wf0DqONnHbom6d+PHHH13yggULWmDIBx5++OGHVpwuPrPteuXKlS6d+8JSuDI+uf6GrY6yq3S6KTOWz17aVSwdyRQSDZ8bEalIGvSmTZui9yxCxNLIaOBXr151d2OzJu/1YwChcnLawxKgQ1Kvr7/+enHo0CG3XyQoXc3LfgmDAA4F//nnnz+kGHRNhvLij8W1Jlw9Q9uZP3++Lbr1rCuQuebVOilQpetDB48//rgFczzoqmISkJe/Byo+pdzaEPQZQN70By7Xs4qxV7SxfPaKPyZfMoUEM1SGRtaXX3659ewTwiX7fKGkk6Oh79mzp+DLIHbUUYPp5WMAlKmRRDO1TnQMYzqzXeqT2y3p1DNmzGgTA/XM7AKFxeYos1a+vsJARCeRo565W0mzUTomuDm7V6djVke5DGhSClXlPfHEE1VJbfFq222RXqBqVeCB9R2UvFFE0KU+EULMSyPqhrxVLyrIH8NnqJx+0pIpJEYZRgtdAgXj3U6hLSM0XN4McAGaFIhN57lMcL1+DMDHPexh3koy06VOUUxr1qxpXfaFbBgsWNpQz1zNCyyfmUIJ2M81Hzt2zF3bi7KiI5Cn7mtxoY3PUvnKcSLVKfLmRx9jUOBmT81Uy/hkuc3Awn3v5Nu1a5erK13KV5ZnEHHJFBINkzuOaJA0vH4coyedgBG4TMH0gzvn7U4CKH0aPPVhbyPYu3evqx9uXtQMhM9MoQRo9FqikEZ+daBLly4V+iRUp09mdUfpv9AMZLwWjx0Qr1+//m/mwJP20gIghT52EYJJmaaBn4Hjm2++qUQNHLLXIM5nuxgc6K++i+HTz5MqnEwhQRCjqtbUmin1QijTfBowjRtbJTXuXnDlPGkkwFKEDWo5bUhLGSleSwftvyje+uo47EOldrQ7aIpRRrqWuGyPC7q09ySeyj7uyca23JQpU9pkpHj8Ou+x1pKzig9Lh33WZr4UUCyfFkfq5yQKiUagPQI0MIokhRJh9GWtiybPbmwkQD0ic3/6T5hZkpxGWr+e1SnU2QUvH3g2vnkxIRxK69fnLSyzAJaJchjusWdV5tiXZBnjXxZIGEUjZav7z3WfOriQBzN5PpAgx2eVcNZgUs8vvviiwPrymf35/Oi+eX9zXgVRp6xa/Drl7Rz8a382lk/hrcW39gFlzyGbAcFjs2FtVAhb+wzByZd9iMLWJy+2IXLYdECDxYf9CnHWVkZx9q7rsjjwytYmZJ+i8pvox9SJT7dk4fMsWVAnOIWRt4z5ZCxoZSs46l31QBx2R7auRAe4sMXp1DYELz+WV9kFWRsgcGCH5fOmMOlqi7ITkt0QvFgHn/Am/PBIWDIClmfiZBhJmGfbNyxO+xzLp+gQffBNGbbPqG7Ep8JldSoY0SL8IT4Fi1/VriyM/xzilaVR0IUyKyONDDj9OlWAGoHyy5e1LXjUqKWQiKMcNRjCVD7OwqjRlMWpHFUQ/nh08B7rqIuy+lEHlCzUMOlEKCAaOOXw49kqI5VNXltnlCM8gsFXXQCrcm166DmWV9qL6PV90eTzqnJJp92QDx7KeEUutgzkCl++I4400WCVgA9rw8DHOHgAp+jF98vw+YR2FIdfp2V1Ectnp3YV4iXEa9SNkeznpHRMIXnrMjIyUvkGLWV5Pq5uLtT38zYhzPQ7dZ00ga8yGoaF12HhkzoO8ZpkD6msIYXieI3P2tWuyUPwqdPyhfqpJZrxZQmkkUDUFbZpimrHwmeMsex+6qmn2gzp2qHShtjUO3LkiNuMxPI7uyyBLIFmSWAgMyREgInA8ePHnSU2Uzi9patLPLyZwKKYNwsoI71ZqKu8jDdLIEugewlE7SF1jzbnyBLIEsgSqJZA1R5oxyVbVcbqonJKlkCWQJZAbxIY2JKtN3JzriyBLIGJLIGskCZy7WbesgTGmQT+B4C9UGezQrbnAAAAAElFTkSuQmCC"></p>
</div>
<div class="specification">
<p>Hence find</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate her distance after 8 minutes. Give your answer in km, correct to 3 decimal places.</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 <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c">
<mi>c</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d">
<mi>d</mi>
</math></span>.</p>
<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 distance she runs in 20 minutes.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>her maximum speed, in ms<sup>ā1</sup>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = 3x\arccos (x)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>3</mn>
<mi>x</mi>
<mi>arccos</mi>
<mo>ā”<!-- ā” --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 1 \leqslant x \leqslant 1">
<mo>ā<!-- ā --></mo>
<mn>1</mn>
<mo>ā©½<!-- ā©½ --></mo>
<mi>x</mi>
<mo>ā©½<!-- ā©½ --></mo>
<mn>1</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> indicating clearly any intercepts with the axes and the coordinates of any local maximum or minimum points.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the range of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve the inequality <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| {3x\arccos (x)} \right| > 1">
<mrow>
<mo>|</mo>
<mrow>
<mn>3</mn>
<mi>x</mi>
<mi>arccos</mi>
<mo>ā”</mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mrow>
<mo>|</mo>
</mrow>
<mo>></mo>
<mn>1</mn>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Jorge is carefully observing the rise in sales of a new app he has created.</p>
<p>The number of sales in the first four months is shown in the table below.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">Jorge believes that the increase is exponential and proposes to model the number of sales <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math> in month <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> with the equation</p>
<p style="text-align: left; padding-left: 30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>=</mo><mi>A</mi><msup><mtext>e</mtext><mrow><mi>r</mi><mi>t</mi></mrow></msup><mo>,</mo><mo> </mo><mi>A</mi><mo>,</mo><mo>ā</mo><mi>r</mi><mo>ā</mo><mi mathvariant="normal">ā</mi></math></p>
</div>
<div class="specification">
<p>Jorge plans to adapt Eulerās method to find an approximate value for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math>.</p>
<p>With a step length of one month the solution to the differential equation can be approximated using Eulerās method where</p>
<p style="padding-left: 30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>ā</mo><mi>N</mi><mfenced><mi>n</mi></mfenced><mo>+</mo><mn>1</mn><mo>Ć</mo><mi>N</mi><mo>'</mo><mfenced><mi>n</mi></mfenced><mo>,</mo><mo> </mo><mi>n</mi><mo>ā</mo><mi mathvariant="normal">ā</mi></math></p>
</div>
<div class="specification">
<p>Jorge decides to take the mean of these values as the approximation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> for his model. He also decides the graph of the model should pass through the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>2</mn><mo>,</mo><mo> </mo><mn>52</mn><mo>)</mo></math>.</p>
</div>
<div class="specification">
<p>The sum of the square residuals for these points for the least squares regression model is approximately <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>.</mo><mn>555</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that Jorgeās model satisfies the differential equation</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>N</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>r</mi><mi>N</mi></math></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>ā</mo><mfrac><mrow><mi>N</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mi>N</mi><mfenced><mi>n</mi></mfenced></mrow><mrow><mi>N</mi><mfenced><mi>n</mi></mfenced></mrow></mfrac></math></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find three approximations for the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation for Jorgeās model.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the sum of the square residuals for Jorgeās model using the values <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>2</mn><mo>,</mo><mo> </mo><mn>3</mn><mo>,</mo><mo> </mo><mn>4</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Comment how well Jorgeās model fits the data.</p>
<div class="marks">[1]</div>
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Give two possible sources of error in the construction of his model.</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msqrt><mi>x</mi></msqrt></math>.</p>
</div>
<div class="specification">
<p>The shape of a piece of metal can be modelled by the region bounded by the functions <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math>, the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis and the line segment <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>[AB]</mtext></math>, as shown in the following diagram. The units on the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> axes are measured in metres.</p>
<p style="text-align: center;"><img 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"></p>
<p>The piecewise function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is defined by</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>{</mo><mtable><mtr><mtd><msqrt><mi>x</mi></msqrt><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>0</mn><mo>.</mo><mn>16</mn></mtd></mtr><mtr><mtd><mn>1</mn><mo>.</mo><mn>25</mn><mi>x</mi><mo>+</mo><mn>0</mn><mo>.</mo><mn>2</mn><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>16</mn><mo><</mo><mi>x</mi><mo>≤</mo><mn>0</mn><mo>.</mo><mn>5</mn></mtd></mtr></mtable></math></p>
<p>The graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> is obtained from the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> by:</p>
<ul>
<li>a stretch scale factor of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac></math> in the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> direction,</li>
<li>followed by a stretch scale factor <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac></math> in the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> direction,</li>
<li>followed by a translation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>2</mn></math> units to the right.</li>
</ul>
<p>Point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> lies on the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> and has coordinates <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>.</mo><mn>5</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>.</mo><mn>825</mn><mo>)</mo></math>. Point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> is the image of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> under the given transformations and has coordinates <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>p</mi><mo>,</mo><mo> </mo><mi>q</mi><mo>)</mo></math>.</p>
</div>
<div class="specification">
<p>The piecewise function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> is given by</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>{</mo><mtable><mtr><mtd><mi>h</mi><mfenced><mi>x</mi></mfenced><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>2</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mi>a</mi></mtd></mtr><mtr><mtd><mn>1</mn><mo>.</mo><mn>25</mn><mi>x</mi><mo>+</mo><mi>b</mi><mo> </mo><mo> </mo></mtd><mtd><mi>a</mi><mo><</mo><mi>x</mi><mo>≤</mo><mi>p</mi></mtd></mtr></mtable></math></p>
</div>
<div class="specification">
<p>The area enclosed by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>, the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis and the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>p</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>0627292</mn><mo> </mo><msup><mtext>m</mtext><mn>2</mn></msup></math> correct to six significant figures.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that the equation of the tangent to the curve at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>.</mo><mn>16</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>.</mo><mn>4</mn></mrow></mfenced></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>25</mn><mi>x</mi><mo>+</mo><mn>0</mn><mo>.</mo><mn>2</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> and the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for<math xmlns="http://www.w3.org/1998/Math/MathML"><mo> </mo><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area enclosed by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>, the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis and the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>5</mn></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area of the shaded region on the diagram.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.ii.</div>
</div>
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