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<h2>HL Paper 2</h2><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><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mi>x</mi><mo>-</mo><mn>12</mn></mrow><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mn>15</mn></mrow></mfrac><mo>,</mo><mo> </mo><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>≠</mo><mfrac><mn>15</mn><mn>2</mn></mfrac></math>.</p>
</div>
<div class="specification">
<p>Find the coordinates where the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> crosses the</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis.</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis.</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>Write down the equation of the vertical asymptote of the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The oblique asymptote of the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> can be written as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>∈</mo><mi mathvariant="normal">ℚ</mi></math>.</p>
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>.</p>
<div class="marks">[4]</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>f</mi></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>30</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>30</mn></math>, clearly indicating the points of intersection with each axis and any asymptotes.</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>Express <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mrow><mi>f</mi><mfenced><mi>x</mi></mfenced></mrow></mfrac></math> in partial fractions.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the exact value of <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mn>0</mn><mn>3</mn></munderover><mfrac><mn>1</mn><mrow><mi>f</mi><mfenced><mi>x</mi></mfenced></mrow></mfrac><mo>d</mo><mi>x</mi></math>, expressing your answer as a single logarithm.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.ii.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Prove the identity <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></mfenced><mn>3</mn></msup><mo>-</mo><mn>3</mn><mi>p</mi><mi>q</mi><mfenced><mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></mfenced><mo>≡</mo><msup><mi>p</mi><mn>3</mn></msup><mo>+</mo><msup><mi>q</mi><mn>3</mn></msup></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>The equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></math> has two real roots, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi></math>.</p>
<p>Consider the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>m</mi><mi>x</mi><mo>+</mo><mi>n</mi><mo>=</mo><mn>0</mn></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>,</mo><mo> </mo><mi>n</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi></math> and which has roots <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><msup><mi>α</mi><mn>3</mn></msup></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><msup><mi>β</mi><mn>3</mn></msup></mfrac></math>.<br>Without solving <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></math>, determine the values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = \frac{{2\,{\text{ln}}\,x + 1}}{{x - 3}}">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mrow>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>3</mn>
</mrow>
</mfrac>
</math></span>, 0 < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> < 3.</p>
</div>
<div class="specification">
<p>Draw a set of axes showing <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span> values between −3 and 3. On these axes</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, or otherwise, find the coordinates of the point of inflexion on the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( x \right)"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( x \right)"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span>, showing clearly any axis intercepts and giving the equations of any asymptotes.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {f^{ - 1}}\left( x \right)"> <mi>y</mi> <mo>=</mo> <mrow> <msup> <mi>f</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span>, showing clearly any axis intercepts and giving the equations of any asymptotes.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, or otherwise, solve the inequality <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) > {f^{ - 1}}\left( x \right)"> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>></mo> <mrow> <msup> <mi>f</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>A 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><mrow><mi>k</mi><msup><mtext>e</mtext><mstyle displaystyle="true"><mfrac><mi>x</mi><mn>2</mn></mfrac></mstyle></msup></mrow><mrow><mn>1</mn><mo>+</mo><msup><mtext>e</mtext><mi>x</mi></msup></mrow></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>≥</mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>∈</mo><msup><mi mathvariant="normal">ℝ</mi><mo>+</mo></msup></math>.</p>
<p>The region enclosed by the graph of <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, the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis and the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>ln</mi><mo> </mo><mn>16</mn></math> is rotated <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 to form a solid of revolution.</p>
</div>
<div class="specification">
<p>Pedro wants to make a small bowl with a volume of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>300</mn><mo> </mo><msup><mtext>cm</mtext><mn>3</mn></msup></math> based on the result from part (a). Pedro’s design is shown in the following diagrams.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">The vertical height of the bowl, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>BO</mtext></math>, is measured along the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis. The radius of the bowl’s top is <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>OA</mtext></math> and the radius of the bowl’s base is <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>BC</mtext></math>. All lengths are measured in <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>cm</mtext></math>.</p>
</div>
<div class="specification">
<p>For design purposes, Pedro investigates how the cross-sectional radius of the bowl changes.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the volume of the solid formed is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>15</mn><msup><mi>k</mi><mn>2</mn></msup><mi mathvariant="normal">π</mi></mrow><mn>34</mn></mfrac></math> cubic units.</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> that satisfies the requirements of Pedro’s design.</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 <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>OA</mtext></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 <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>BC</mtext></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By sketching the graph of a suitable derivative of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math>, find where the cross-sectional radius of the bowl is decreasing most rapidly.</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>State the cross-sectional radius of the bowl at this point.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>A continuous random variable <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> has a probability density function given by</p>
<p style="padding-left: 180px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced open="{" close><mtable><mtr><mtd><mtext>arccos</mtext><mo> </mo><mi>x</mi><mo> </mo></mtd><mtd><mn>0</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mtext>otherwise</mtext></mtd></mtr></mtable></mfenced></math></p>
<p>The median of this distribution is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</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>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mfenced><mrow><mfenced open="|" close="|"><mrow><mi>X</mi><mo>-</mo><mi>m</mi></mrow></mfenced><mo>≤</mo><mi>a</mi></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>3</mn></math>, determine the value 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">b.</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><msqrt><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></msqrt></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>2</mn></math>.</p>
</div>
<div class="specification">
<p>The curve <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> is rotated <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>π</mi></math> about the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis to form a solid of revolution that is used to model a water container.</p>
</div>
<div class="specification">
<p>At <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math>, the container is empty. Water is then added to the container at a constant rate of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>4</mn><mo> </mo><msup><mtext>m</mtext><mn>3</mn></msup><mo> </mo><msup><mtext>s</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch 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>, clearly indicating the coordinates of the endpoints.</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 inverse function of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><msqrt><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></msqrt></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the domain and range of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the volume, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo> </mo><msup><mtext>m</mtext><mn>3</mn></msup></math>, of water in the container when it is filled to a height of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math> metres is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><mi>π</mi><mfenced><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><msup><mi>h</mi><mn>3</mn></msup><mo>+</mo><mi>h</mi></mrow></mfenced></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, determine the maximum volume of the container.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the time it takes to fill the container to its maximum volume.</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>Find the rate of change of the height of the water when the container is filled to half its maximum volume.</p>
<div class="marks">[6]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>The voltage <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v">
<mi>v</mi>
</math></span> in a circuit is given by the equation</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v\left( t \right) = 3\,{\text{sin}}\left( {100\pi t} \right)">
<mi>v</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>100</mn>
<mi>π<!-- π --></mi>
<mi>t</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t \geqslant 0">
<mi>t</mi>
<mo>⩾<!-- ⩾ --></mo>
<mn>0</mn>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> is measured in seconds.</p>
</div>
<div class="specification">
<p>The current <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="i">
<mi>i</mi>
</math></span> in this circuit is given by the equation</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="i\left( t \right) = 2\,{\text{sin}}\left( {100\pi \left( {t + 0.003} \right)} \right)">
<mi>i</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>100</mn>
<mi>π<!-- π --></mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>+</mo>
<mn>0.003</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>The power <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span> in this circuit is given by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\left( t \right) = v\left( t \right) \times i\left( t \right)">
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>v</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>×<!-- × --></mo>
<mi>i</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>The average power <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{p_{av}}">
<mrow>
<msub>
<mi>p</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
</mrow>
</msub>
</mrow>
</math></span> in this circuit from <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 0">
<mi>t</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = T">
<mi>t</mi>
<mo>=</mo>
<mi>T</mi>
</math></span> is given by the equation</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{p_{av}}\left( T \right) = \frac{1}{T}\int_0^T {p\left( t \right){\text{d}}t} ">
<mrow>
<msub>
<mi>p</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mi>T</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mi>T</mi>
</mfrac>
<msubsup>
<mo>∫<!-- ∫ --></mo>
<mn>0</mn>
<mi>T</mi>
</msubsup>
<mrow>
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="T > 0">
<mi>T</mi>
<mo>></mo>
<mn>0</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the maximum and minimum value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v">
<mi>v</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down two transformations that will transform the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = v\left( t \right)">
<mi>y</mi>
<mo>=</mo>
<mi>v</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> onto the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = i\left( t \right)">
<mi>y</mi>
<mo>=</mo>
<mi>i</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = p\left( t \right)">
<mi>y</mi>
<mo>=</mo>
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> for 0 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> ≤ 0.02 , showing clearly the coordinates of the first maximum and the first minimum.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the total time in the interval 0 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> ≤ 0.02 for which <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\left( t \right)">
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> ≥ 3.</p>
<p> </p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{p_{av}}">
<mrow>
<msub>
<mi>p</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
</mrow>
</msub>
</mrow>
</math></span>(0.007).</p>
<p> </p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>With reference to your graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = p\left( t \right)">
<mi>y</mi>
<mo>=</mo>
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> explain why <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{p_{av}}\left( T \right)">
<mrow>
<msub>
<mi>p</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mi>T</mi>
<mo>)</mo>
</mrow>
</math></span> > 0 for all <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="T">
<mi>T</mi>
</math></span> > 0.</p>
<p> </p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\left( t \right)">
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> can be written as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\left( t \right) = a\,{\text{sin}}\left( {b\left( {t - c} \right)} \right) + d">
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>a</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>b</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>−</mo>
<mi>c</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>d</mi>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c">
<mi>c</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d">
<mi>d</mi>
</math></span> > 0, use your graph to find the values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c">
<mi>c</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d">
<mi>d</mi>
</math></span>.</p>
<p> </p>
<div class="marks">[6]</div>
<div class="question_part_label">g.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the rectangle OABC such that AB = OC = 10 and BC = OA = 1 , with the points P , Q and R placed on the line OC such that OP = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span>, OQ = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q">
<mi>q</mi>
</math></span> and OR = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span>, such that 0 < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span> < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q">
<mi>q</mi>
</math></span> < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span> < 10.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\theta _p}">
<mrow>
<msub>
<mi>θ<!-- θ --></mi>
<mi>p</mi>
</msub>
</mrow>
</math></span> be the angle APO, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\theta _q}">
<mrow>
<msub>
<mi>θ<!-- θ --></mi>
<mi>q</mi>
</msub>
</mrow>
</math></span> be the angle AQO and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\theta _r}">
<mrow>
<msub>
<mi>θ<!-- θ --></mi>
<mi>r</mi>
</msub>
</mrow>
</math></span> be the angle ARO.</p>
</div>
<div class="specification">
<p>Consider the case when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\theta _p} = {\theta _q} + {\theta _r}">
<mrow>
<msub>
<mi>θ<!-- θ --></mi>
<mi>p</mi>
</msub>
</mrow>
<mo>=</mo>
<mrow>
<msub>
<mi>θ<!-- θ --></mi>
<mi>q</mi>
</msub>
</mrow>
<mo>+</mo>
<mrow>
<msub>
<mi>θ<!-- θ --></mi>
<mi>r</mi>
</msub>
</mrow>
</math></span> and QR = 1.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\theta _p}"> <mrow> <msub> <mi>θ</mi> <mi>p</mi> </msub> </mrow> </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">[3]</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="p = \frac{{{q^2} + q - 1}}{{2q + 1}}"> <mi>p</mi> <mo>=</mo> <mfrac> <mrow> <mrow> <msup> <mi>q</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mi>q</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <mi>q</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </math></span>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By sketching the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span> as a function of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q"> <mi>q</mi> </math></span>, determine the range of values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span> for which there are possible values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q"> <mi>q</mi> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cot}}\,2\theta = \frac{{1 - {\text{ta}}{{\text{n}}^2}\,\theta }}{{2\,{\text{tan}}\,\theta }}">
<mrow>
<mtext>cot</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>θ</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<mtext>ta</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</mrow>
<mrow>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</mrow>
</mfrac>
</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>Verify that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = {\text{tan}}\,\theta ">
<mi>x</mi>
<mo>=</mo>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = - \,{\text{cot}}\,\theta ">
<mi>x</mi>
<mo>=</mo>
<mo>−</mo>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cot</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</math></span> satisfy the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} + \left( {2\,{\text{cot}}\,2\theta } \right)x - 1 = 0">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cot</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>θ</mi>
</mrow>
<mo>)</mo>
</mrow>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
<mo>=</mo>
<mn>0</mn>
</math></span>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, or otherwise, show that the exact value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\frac{\pi }{{12}} = 2 - \sqrt 3 ">
<mrow>
<mtext>tan</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mn>2</mn>
<mo>−</mo>
<msqrt>
<mn>3</mn>
</msqrt>
</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>Using the results from parts (b) and (c) find the exact value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\frac{\pi }{{24}} - {\text{cot}}\frac{\pi }{{24}}">
<mrow>
<mtext>tan</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>24</mn>
</mrow>
</mfrac>
<mo>−</mo>
<mrow>
<mtext>cot</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>24</mn>
</mrow>
</mfrac>
</math></span>.</p>
<p>Give your answer in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a + b\sqrt 3 ">
<mi>a</mi>
<mo>+</mo>
<mi>b</mi>
<msqrt>
<mn>3</mn>
</msqrt>
</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 \in \mathbb{Z}">
<mi>b</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
</math></span>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>A 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><mtext>arcsin</mtext><mfenced><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced><mo>,</mo><mo> </mo><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
</div>
<div class="specification">
<p>A function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mtext>arcsin</mtext><mfenced><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced><mo>,</mo><mo> </mo><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><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>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is an even function.</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>By considering limits, show that the graph of <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> has a horizontal asymptote and state its equation.</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>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><msqrt><msup><mi>x</mi><mn>2</mn></msup></msqrt><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfrac></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>≠</mo><mn>0</mn></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By using the expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced></math> and the result <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><msup><mi>x</mi><mn>2</mn></msup></msqrt><mo>=</mo><mfenced open="|" close="|"><mi>x</mi></mfenced></math>, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is decreasing for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo><</mo><mn>0</mn></math>.</p>
<p> </p>
<div class="marks">[3]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo></math>, justifying your answer.</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the domain of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo></math>, clearly indicating any asymptotes with their equations and stating the values of any axes intercepts.</p>
<div class="marks">[3]</div>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p>The height of water, in metres, in Dungeness harbour is modelled by the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mi>a</mi><mo> </mo><mi>sin</mi><mo>(</mo><mi>b</mi><mo>(</mo><mi>t</mi><mo>-</mo><mi>c</mi><mo>)</mo><mo>)</mo><mo>+</mo><mi>d</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> is the number of hours after midnight, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>,</mo><mo> </mo><mi>c</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math> are constants, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>></mo><mn>0</mn><mo>,</mo><mo> </mo><mi>b</mi><mo>></mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>></mo><mn>0</mn></math>.</p>
<p>The following graph shows the height of the water for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>13</mn></math> hours, starting at midnight.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p>The first high tide occurs at <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>04</mn><mo>:</mo><mn>30</mn></math> and the next high tide occurs <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn></math> hours later. Throughout the day, the height of the water fluctuates between <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>2</mn><mo> </mo><mtext>m</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>.</mo><mn>8</mn><mo> </mo><mtext>m</mtext></math>.</p>
<p>All heights are given correct to one decimal place.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></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>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">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>d</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the smallest possible value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</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>Find the height of the water at <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn><mo>:</mo><mn>00</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>Determine the number of hours, over a 24-hour period, for which the tide is higher than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math> metres.</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>A fisherman notes that the water height at nearby Folkestone harbour follows the same sinusoidal pattern as that of Dungeness harbour, with the exception that high tides (and low tides) occur <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>50</mn></math> minutes earlier than at Dungeness.</p>
<p>Find a suitable equation that may be used to model the tidal height of water at Folkestone harbour.</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.</div>
</div>
<br><hr><br><div class="specification">
<p>It is given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = 3{x^4} + a{x^3} + b{x^2} - 7x - 4">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>a</mi>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>b</mi>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>7</mn>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>4</mn>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span> are positive integers.</p>
</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="{x^2} - 1">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</math></span> is a factor 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> find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> and 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">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Factorize <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> into a product of linear factors.</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>Using your graph state the range of values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c">
<mi>c</mi>
</math></span> for which <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = c">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>c</mi>
</math></span> has exactly two distinct real roots.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</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> ≤ 9.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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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>−</mo> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>−</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> ≤ 9.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = - 1 + \ln \left( {\sqrt {{x^2} - 1} } \right)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>+</mo>
<mi>ln</mi>
<mo><!-- --></mo>
<mrow>
<mo>(</mo>
<mrow>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
</div>
<div class="specification">
<p>The function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = - 1 + \ln \left( {\sqrt {{x^2} - 1} } \right),{\text{ }}x \in D">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>+</mo>
<mi>ln</mi>
<mo><!-- --></mo>
<mrow>
<mo>(</mo>
<mrow>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mi>D</mi>
</math></span></p>
</div>
<div class="specification">
<p>The function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g">
<mi>g</mi>
</math></span> is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g(x) = - 1 + \ln \left( {\sqrt {{x^2} - 1} } \right),{\text{ }}x \in \left] {1,{\text{ }}\infty } \right[">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>+</mo>
<mi>ln</mi>
<mo><!-- --></mo>
<mrow>
<mo>(</mo>
<mrow>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mo>]</mo>
<mrow>
<mn>1</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
<mo>[</mo>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the largest possible domain <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="D">
<mi>D</mi>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> to be a function.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f(x)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span> showing clearly the equations of asymptotes and the coordinates of any intercepts with the axes.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain why <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is an even function.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain why the inverse function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f^{ - 1}}">
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span> does not exist.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the inverse function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{g^{ - 1}}">
<mrow>
<msup>
<mi>g</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span> and state its domain.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'(x)">
<msup>
<mi>g</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, show that there are no solutions to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'(x) = 0">
<msup>
<mi>g</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</math></span>;</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, show that there are no solutions to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="({g^{ - 1}})'(x) = 0">
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>g</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.ii.</div>
</div>
<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><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow></mfrac></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>≠</mo><mi>p</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>≠</mo><mi>q</mi></math>.</p>
</div>
<div class="specification">
<p>The graph of <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> has exactly one point of inflexion.</p>
</div>
<div class="specification">
<p>The function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfrac></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>≠</mo><mo>-</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></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>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">a.</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"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced></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>Find the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-coordinate of the point of inflexion.</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>Sketch the graph of <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> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>3</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>3</mn></math>, showing the values of any axes intercepts, the coordinates of any local maxima and local minima, and giving the equations of any asymptotes.</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equations of all the asymptotes on the graph of <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>.</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>By considering the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>-</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>, or otherwise, solve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo><</mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">f.</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><!-- --></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>−<!-- − --></mo>
<mn>1</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</mi>
<mo>⩽<!-- ⩽ --></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></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>The population, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math>, of a particular species of marsupial on a small remote island can be modelled by the logistic differential equation</p>
<p style="padding-left: 180px;"><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><mi>k</mi><mi>P</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>P</mi><mi>N</mi></mfrac></mrow></mfenced></math></p>
<p>where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> is the time measured in years and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>,</mo><mo> </mo><mi>N</mi></math> are positive constants.</p>
<p>The constant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math> represents the maximum population of this species of marsupial that the island can sustain indefinitely.</p>
</div>
<div class="specification">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>0</mn></msub></math> be the initial population of marsupials.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>In the context of the population model, interpret the meaning of <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></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 <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>P</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><msup><mi>k</mi><mn>2</mn></msup><mi>P</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>P</mi><mi>N</mi></mfrac></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mrow><mn>2</mn><mi>P</mi></mrow><mi>N</mi></mfrac></mrow></mfenced></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that the population of marsupials will increase at its maximum rate when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mfrac><mi>N</mi><mn>2</mn></mfrac></math>. Justify your answer.</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>Hence determine the maximum value of <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></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>N</mi></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>By solving the logistic differential equation, show that its solution can be expressed in the form</p>
<p style="padding-left:150px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mi>t</mi><mo>=</mo><mi>ln</mi><mfrac><mi>P</mi><msub><mi>P</mi><mn>0</mn></msub></mfrac><mfenced><mfrac><mrow><mi>N</mi><mo>-</mo><msub><mi>P</mi><mn>0</mn></msub></mrow><mrow><mi>N</mi><mo>-</mo><mi>P</mi></mrow></mfrac></mfenced></math>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>After <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> years, the population of marsupials is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><msub><mi>P</mi><mn>0</mn></msub></math>. It is known that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>=</mo><mn>4</mn><msub><mi>P</mi><mn>0</mn></msub></math>.</p>
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> for this population model.</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</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><mn>2</mn><mi>x</mi></msup><mo>-</mo><mfrac><mn>1</mn><msup><mn>2</mn><mi>x</mi></msup></mfrac><mo>,</mo><mo> </mo><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
</div>
<div class="specification">
<p>The function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>≠</mo><mo>-</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>≠</mo><mn>3</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is an odd 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>Solve the inequality <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>≥</mo><mi>g</mi><mfenced><mi>x</mi></mfenced></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A scientist conducted a nine-week experiment on two plants, <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>, of the same species. He wanted to determine the effect of using a new plant fertilizer. Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> was given fertilizer regularly, while Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> was not.</p>
<p>The scientist found that the height of Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>,</mo><mo> </mo><msub><mi>h</mi><mi>A</mi></msub><mo> </mo><mtext>cm</mtext></math>, at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> weeks can be modelled by the function <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>h</mi><mi>A</mi></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mi>sin</mi><mo>(</mo><mn>2</mn><mi>t</mi><mo>+</mo><mn>6</mn><mo>)</mo><mo>+</mo><mn>9</mn><mi>t</mi><mo>+</mo><mn>27</mn></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mn>9</mn></math>.</p>
<p>The scientist found that the height of Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>,</mo><mo> </mo><msub><mi>h</mi><mi>B</mi></msub><mo> </mo><mtext>cm</mtext></math>, at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> weeks can be modelled by the function <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>h</mi><mi>B</mi></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mn>8</mn><mi>t</mi><mo>+</mo><mn>32</mn></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mn>9</mn></math>.</p>
</div>
<div class="specification">
<p>Use the scientist’s models to find the initial height of</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</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>Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> correct to three significant figures.</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 values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> when <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>h</mi><mi>A</mi></msub><mfenced><mi>t</mi></mfenced><mo>=</mo><msub><mi>h</mi><mi>B</mi></msub><mfenced><mi>t</mi></mfenced></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>For <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>></mo><mn>6</mn></math>, prove that Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> was always taller than Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>For <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mn>9</mn></math>, find the total amount of time when the rate of growth of Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> was greater than the rate of growth of Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</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>π<!-- π --></mi>
<mn>4</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>cot</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>π<!-- π --></mi>
<mn>4</mn>
</mfrac>
<mo>−<!-- − --></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>α<!-- α --></mi>
</math></span>, <em>β</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>−</mo> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mn>8</mn> </mfrac> <mo>⩽</mo> <mi>x</mi> <mo>⩽</mo> <mfrac> <mi>π</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>−</mo> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> <mo>⩽</mo> <mi>x</mi> <mo>⩽</mo> <mfrac> <mi>π</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>−</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> ≤ 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>α</mi> </math></span> and <em>β</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>α</mi> </math></span> + <em>β</em> < −2.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P\left( x \right) = 2{x^4} - 15{x^3} + a{x^2} + bx + c">
<mi>P</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>2</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>15</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<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>, 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 \in \mathbb{R}">
<mi>c</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>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {x - 5} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>5</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> is a factor of <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>, find a relationship between <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> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c">
<mi>c</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>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {x - 5} \right)^2}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>5</mn>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span> is a factor of <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>, write down the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P'\left( 5 \right)">
<msup>
<mi>P</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mn>5</mn>
<mo>)</mo>
</mrow>
</math></span>.</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>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {x - 5} \right)^2}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>5</mn>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span> is a factor of <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>, and that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = 2">
<mi>a</mi>
<mo>=</mo>
<mn>2</mn>
</math></span>, find the values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c">
<mi>c</mi>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^5} - 3{x^4} + m{x^3} + n{x^2} + px + q = 0">
<mrow>
<msup>
<mi>x</mi>
<mn>5</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>m</mi>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>n</mi>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>p</mi>
<mi>x</mi>
<mo>+</mo>
<mi>q</mi>
<mo>=</mo>
<mn>0</mn>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m">
<mi>m</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q \in \mathbb{R}">
<mi>q</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>.</p>
<p>The equation has three distinct real roots which can be written as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_2}\,a">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>a</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_2}\,b">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>b</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_2}\,c">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>c</mi>
</math></span>.</p>
<p>The equation also has two imaginary roots, one of which is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d{\text{i}}">
<mi>d</mi>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d \in \mathbb{R}">
<mi>d</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>The values <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>, and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c">
<mi>c</mi>
</math></span> are consecutive terms in a geometric sequence.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="abc = 8"> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mo>=</mo> <mn>8</mn> </math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that one of the real roots is equal to 1.</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>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q = 8{d^2}"> <mi>q</mi> <mo>=</mo> <mn>8</mn> <mrow> <msup> <mi>d</mi> <mn>2</mn> </msup> </mrow> </math></span>, find the other two real roots.</p>
<div class="marks">[9]</div>
<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="y = f\left( x \right)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 3 \leqslant x \leqslant 5">
<mo>−<!-- − --></mo>
<mn>3</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</mi>
<mo>⩽<!-- ⩽ --></mo>
<mn>5</mn>
</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>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {f \circ f} \right)\left( 1 \right)"> <mrow> <mo>(</mo> <mrow> <mi>f</mi> <mo>∘</mo> <mi>f</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mn>1</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>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f^{ - 1}}\left( a \right) = 3"> <mrow> <msup> <mi>f</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>3</mn> </math></span>, determine 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">[2]</div>
<div class="question_part_label">b.</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="g\left( x \right) = 2f\left( {x - 1} \right)"> <mi>g</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>2</mn> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </math></span>, find the domain and range of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g"> <mi>g</mi> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the set of values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k"> <mi>k</mi> </math></span> that satisfy the inequality <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{k^2} - k - 12 < 0"> <mrow> <msup> <mi>k</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mi>k</mi> <mo>−</mo> <mn>12</mn> <mo><</mo> <mn>0</mn> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The triangle ABC is shown in the following diagram. Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos B < \frac{1}{4}"> <mi>cos</mi> <mo></mo> <mi>B</mi> <mo><</mo> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </math></span>, find the range of possible values for AB.</p>
<p><img src="images/Schermafbeelding_2017-08-09_om_18.13.24.png" alt="M17/5/MATHL/HP2/ENG/TZ2/04.b"></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\left( x \right) = \frac{{ax + 1}}{{bx + c}}">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mi>a</mi>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mrow>
<mi>b</mi>
<mi>x</mi>
<mo>+</mo>
<mi>c</mi>
</mrow>
</mfrac>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \ne - \frac{c}{b}">
<mi>x</mi>
<mo>≠<!-- ≠ --></mo>
<mo>−<!-- − --></mo>
<mfrac>
<mi>c</mi>
<mi>b</mi>
</mfrac>
</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 \in \mathbb{Z}">
<mi>c</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
</math></span>.</p>
<p>The following graph shows the curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {\left( {f\left( x \right)} \right)^2}">
<mi>y</mi>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span>. It has asymptotes at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = p">
<mi>x</mi>
<mo>=</mo>
<mi>p</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = q">
<mi>y</mi>
<mo>=</mo>
<mi>q</mi>
</math></span> and meets the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis at A.</p>
<p style="text-align: center;"><img 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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>On the following axes, sketch the two possible graphs 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> giving the equations of any asymptotes in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q"> <mi>q</mi> </math></span>.</p>
<p><img src="data:image/png;base64,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"></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>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p = \frac{4}{3}"> <mi>p</mi> <mo>=</mo> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q = \frac{4}{9}"> <mi>q</mi> <mo>=</mo> <mfrac> <mn>4</mn> <mn>9</mn> </mfrac> </math></span> and A has coordinates <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( { - \frac{1}{2},\,\,0} \right)"> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </math></span>, determine the possible sets of values for <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> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c"> <mi>c</mi> </math></span>.</p>
<div class="marks">[4]</div>
<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> has a derivative given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><mrow><mi>x</mi><mfenced><mrow><mi>k</mi><mo>-</mo><mi>x</mi></mrow></mfenced></mrow></mfrac><mo>,</mo><mo> </mo><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>≠</mo><mi>o</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>≠</mo><mi>k</mi></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> is a positive constant.</p>
</div>
<div class="specification">
<p>Consider <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math>, the population of a colony of ants, which has an initial value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1200</mn></math>.</p>
<p>The rate of change of the population can be modelled by the differential equation <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><mfrac><mrow><mi>P</mi><mfenced><mrow><mi>k</mi><mo>-</mo><mi>P</mi></mrow></mfenced></mrow><mrow><mn>5</mn><mi>k</mi></mrow></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> is the time measured in days, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>≥</mo><mn>0</mn></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> is the upper bound for the population.</p>
</div>
<div class="specification">
<p>At <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>10</mn></math> the population of the colony has doubled in size from its initial value.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>′</mo><mo>(</mo><mi>x</mi><mo>)</mo></math> can be written in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>a</mi><mi>x</mi></mfrac><mo>+</mo><mfrac><mi>b</mi><mrow><mi>k</mi><mo>-</mo><mi>x</mi></mrow></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>. Find <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> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></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>By solving the differential equation, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mfrac><mrow><mn>1200</mn><mi>k</mi></mrow><mrow><mfenced><mrow><mi>k</mi><mo>-</mo><mn>1200</mn></mrow></mfenced><msup><mtext>e</mtext><mrow><mo>-</mo><mstyle displaystyle="true"><mfrac><mi>t</mi><mn>5</mn></mfrac></mstyle></mrow></msup><mo>+</mo><mn>1200</mn></mrow></mfrac></math>.</p>
<div class="marks">[8]</div>
<div class="question_part_label">c.</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>, giving your answer correct to four significant figures.</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 value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> when the rate of change of the population is at its maximum.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>Two airplanes, <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>, have position vectors with respect to an origin <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math> given respectively by</p>
<p style="padding-left: 180px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">r</mi><mtext mathvariant="bold-italic">A</mtext></msub><mo>=</mo><mfenced><mtable><mtr><mtd><mn>19</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>t</mi><mfenced><mtable><mtr><mtd><mo>-</mo><mn>6</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mn>4</mn></mtd></mtr></mtable></mfenced></math></p>
<p style="padding-left: 180px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">r</mi><mi mathvariant="bold-italic">B</mi></msub><mo>=</mo><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>12</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>t</mi><mfenced><mtable><mtr><mtd><mn>4</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr></mtable></mfenced></math></p>
<p>where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> represents the time in minutes and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mn>2</mn><mo>.</mo><mn>5</mn></math>.</p>
<p>Entries in each column vector give the displacement east of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math>, the displacement north of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math> and the distance above sea level, all measured in kilometres.</p>
</div>
<div class="specification">
<p>The two airplanes’ lines of flight cross at point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the three-figure bearing on which airplane <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> is travelling.</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 airplane <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> travels at a greater speed than airplane <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</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 the acute angle between the two airplanes’ lines of flight. Give your answer in degrees.</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 coordinates of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>.</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>Determine the length of time between the first airplane arriving at <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> and the second airplane arriving at <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>(</mo><mi>t</mi><mo>)</mo></math> represent the distance between airplane <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and airplane <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mn>2</mn><mo>.</mo><mn>5</mn></math>.</p>
<p>Find the minimum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>(</mo><mi>t</mi><mo>)</mo></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>The number of bananas that Lucca eats during any particular day follows a Poisson distribution with mean 0.2.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that Lucca eats at least one banana in a particular day.</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 expected number of weeks in the year in which Lucca eats no bananas.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A continuous random variable <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> has the probability density function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> given by</p>
<p style="padding-left: 210px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced open="{" close><mtable columnalign="left"><mtr><mtd><mfrac><mi>x</mi><msqrt><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>k</mi></mrow></mfenced><mn>3</mn></msup></msqrt></mfrac><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mn>0</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>4</mn></mtd></mtr><mtr><mtd><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mn>0</mn><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mtext>otherwise</mtext></mtd></mtr></mtable></mfenced></math></p>
<p>where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>∈</mo><msup><mi mathvariant="normal">ℝ</mi><mo>+</mo></msup></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>16</mn><mo>+</mo><mi>k</mi></msqrt><mo>-</mo><msqrt><mi>k</mi></msqrt><mo>=</mo><msqrt><mi>k</mi></msqrt><msqrt><mn>16</mn><mo>+</mo><mi>k</mi></msqrt></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>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>The polynomial <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^4} + p{x^3} + q{x^2} + rx + 6">
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>p</mi>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>q</mi>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>r</mi>
<mi>x</mi>
<mo>+</mo>
<mn>6</mn>
</math></span> is exactly divisible by each of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {x - 1} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {x - 2} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {x - 3} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>3</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<p>Find the values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q">
<mi>q</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span>.</p>
</div>
<br><hr><br><div class="specification">
<p>The function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = {\text{sec}}\,x + 2">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mtext>sec</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 \leqslant x < \frac{\pi }{2}">
<mn>0</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</mi>
<mo><</mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>2</mn>
</mfrac>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down 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">[1]</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><mrow><mo>-</mo><mn>1</mn></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></math></span>, stating its domain.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Consider the graphs of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \frac{{{x^2}}}{{x - 3}}">
<mi>y</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mfrac>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = m\left( {x + 3} \right)">
<mi>y</mi>
<mo>=</mo>
<mi>m</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>3</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m \in \mathbb{R}">
<mi>m</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>.</p>
<p>Find the set of values for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m">
<mi>m</mi>
</math></span> such that the two graphs have no intersection points.</p>
</div>
<br><hr><br><div class="specification">
<p>Consider the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mfenced><mrow><mi>k</mi><mo>+</mo><mn>3</mn></mrow></mfenced><mi>x</mi><mo>+</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>9</mn><mo>=</mo><mn>0</mn></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down an expression for the product of the roots, in terms 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>Hence or otherwise, determine the values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> such that the equation has one positive and one negative real root.</p>
<div class="marks">[3]</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><!-- --></mo>
<mi>x</mi>
</mrow>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>0</mn>
<mo><</mo>
<mi>x</mi>
<mo><</mo>
<mi>π<!-- π --></mi>
</math></span>.</p>
</div>
<div class="specification">
<p>Consider the region bounded by the curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f(x)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span>, the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis and the lines <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{\pi }{6},{\text{ }}x = \frac{\pi }{3}">
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>6</mn>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>3</mn>
</mfrac>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-coordinate of the minimum point on the curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f(x)"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> satisfies the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\tan x = 2x"> <mi>tan</mi> <mo></mo> <mi>x</mi> <mo>=</mo> <mn>2</mn> <mi>x</mi> </math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.i.</div>
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<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>
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<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>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the coordinates of the point on the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span> where the normal to the graph is parallel to the line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = - x"> <mi>y</mi> <mo>=</mo> <mo>−</mo> <mi>x</mi> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>This region is now rotated through <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\pi "> <mn>2</mn> <mi>π</mi> </math></span> radians about the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-axis. Find the volume of revolution.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
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