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<h2>SL Paper 2</h2><div class="specification">
<p>An arithmetic sequence has first term <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>60</mn></math> and common difference <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><mo>.</mo><mn>5</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>th term of the sequence is zero, 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">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mi>n</mi></msub></math> denote the sum of the first <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> terms of the sequence.</p>
<p>Find the maximum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mi>n</mi></msub></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The temperature <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo> </mo><mo>°</mo><mi mathvariant="normal">C</mi></math> of water <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> minutes after being poured into a cup can be modelled by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>=</mo><msub><mi>T</mi><mn>0</mn></msub><msup><mi mathvariant="normal">e</mi><mrow><mo>-</mo><mi>k</mi><mi>t</mi></mrow></msup></math> where <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"><msub><mi>T</mi><mn>0</mn></msub><mo>,</mo><mo> </mo><mi>k</mi></math> are positive constants.</p>
<p>The water is initially boiling at <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn><mo> </mo><mo>°</mo><mtext>C</mtext></math>. When <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>10</mn></math>, the temperature of the water is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>70</mn><mo> </mo><mo>°</mo><mtext>C</mtext></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"><msub><mi>T</mi><mn>0</mn></msub><mo>=</mo><mn>100</mn></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"><mi>k</mi><mo>=</mo><mfrac><mn>1</mn><mn>10</mn></mfrac><mi>ln</mi><mfrac><mn>10</mn><mn>7</mn></mfrac></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the temperature of the water when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>15</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math> versus <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>, clearly indicating any asymptotes with their equations and stating the coordinates of any points of intersection with the axes.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the time taken for the water to have a temperature of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>50</mn><mo> </mo><mo>°</mo><mtext>C</mtext></math>. Give your answer correct to the nearest second.</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>The model for the temperature of the water can also be expressed in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>=</mo><msub><mi>T</mi><mn>0</mn></msub><msup><mi>a</mi><mfrac><mi>t</mi><mn>10</mn></mfrac></msup></math> for <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>a</mi></math> is a positive constant.</p>
<p>Find the exact value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the curves <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mn>1</mn><mo>-</mo><msqrt><mn>1</mn><mo>+</mo><mn>4</mn><msup><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mn>2</mn></msup></msqrt></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mi mathvariant="normal">π</mi><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>0</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-coordinates of the points of intersection of the two curves.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math>, of the region enclosed by the two curves.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is given by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = (2x + 2)(5 - {x^2})">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mn>5</mn>
<mo>−<!-- − --></mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</math></span>.</p>
</div>
<div class="specification">
<p>The graph of the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g(x) = {5^x} + 6x - 6">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<msup>
<mn>5</mn>
<mi>x</mi>
</msup>
</mrow>
<mo>+</mo>
<mn>6</mn>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>6</mn>
</math></span> intersects the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Expand the expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f’(x)">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><strong>Draw </strong>the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 3 \leqslant x \leqslant 3">
<mo>−</mo>
<mn>3</mn>
<mo>⩽</mo>
<mi>x</mi>
<mo>⩽</mo>
<mn>3</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 40 \leqslant y \leqslant 20">
<mo>−</mo>
<mn>40</mn>
<mo>⩽</mo>
<mi>y</mi>
<mo>⩽</mo>
<mn>20</mn>
</math></span>. Use a scale of 2 cm to represent 1 unit on the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis and 1 cm to represent 5 units on the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-axis.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the coordinates of the point of intersection.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</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{1}{3}{x^3} + \frac{3}{4}{x^2} - x - 1">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mfrac>
<mn>3</mn>
<mn>4</mn>
</mfrac>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</math></span>.</p>
</div>
<div class="specification">
<p>The function has one local maximum 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 one local minimum at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = q">
<mi>x</mi>
<mo>=</mo>
<mi>q</mi>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-intercept of 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">[1]</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> for −3 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> ≤ 3 and −4 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span> ≤ 12.</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>Determine the range of <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> for <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="x">
<mi>x</mi>
</math></span> ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q">
<mi>q</mi>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">h.</div>
</div>
<br><hr><br><div class="specification">
<p>The following diagram shows a circle with centre <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math> and radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math>.</p>
<p style="text-align: center;"><img 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"></p>
<p style="text-align: left;">Points <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> lie on the circumference of the circle.</p>
<p style="text-align: left;">Chord <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mtext>AB</mtext></mfenced></math> has length <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mover><mtext>O</mtext><mo>^</mo></mover><mtext>B</mtext><mo>=</mo><mi>θ</mi></math> radians.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that arc <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>APB</mtext></math> has length <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mi mathvariant="normal">π</mi><mo>-</mo><mn>3</mn><mi>θ</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mo>=</mo><msqrt><mn>18</mn><mo>-</mo><mn>18</mn><mo> </mo><mi>cos</mi><mo> </mo><mi>θ</mi></msqrt></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>Arc <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>APB</mtext></math> is twice the length of chord <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mtext>AB</mtext></mfenced></math>.</p>
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Points <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> lie on opposite banks of a river, such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>AP</mtext></math> is the shortest distance across the river. Point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> represents the centre of a city which is located on the riverbank. <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>PB</mtext><mo>=</mo><mn>215</mn><mo> </mo><mtext>km</mtext></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>AP</mtext><mo>=</mo><mn>65</mn><mo> </mo><mtext>km</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mover><mtext>P</mtext><mo>^</mo></mover><mtext>B</mtext><mo>=</mo><mn>90</mn><mo>°</mo></math>.</p>
<p>The following diagram shows this information.</p>
<p><img src="data:image/png;base64,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"></p>
<p>A boat travels at an average speed of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>42</mn><mo> </mo><msup><mtext>km h</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>. A bus travels along the straight road between <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> at an average speed of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>84</mn><mo> </mo><msup><mtext>km h</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>.</p>
</div>
<div class="specification">
<p>Find the travel time, in hours, from <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> given that</p>
</div>
<div class="specification">
<p>There is a point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>D</mtext></math>, which lies on the road from <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math>, such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>BD</mtext><mo>=</mo><mi>x</mi><mo> </mo><mtext>km</mtext></math>. The boat travels from <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>D</mtext></math>, and the bus travels from <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>D</mtext></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math>.</p>
</div>
<div class="specification">
<p>An excursion involves renting the boat and the bus. The cost to rent the boat is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>$</mo><mo> </mo><mn>200</mn></math> per hour, and the cost to rent the bus is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>$</mo><mo> </mo><mn>150</mn></math> per hour.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the boat is taken from <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>, and the bus from <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the boat travels directly to <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression, in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> for the travel time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>, from <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math>, passing through <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>D</mtext></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> so that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math> is a minimum.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the minimum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the new value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> so that the total cost <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> to travel from <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> via <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>D</mtext></math> is a minimum.</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>Write down the minimum total cost for this journey.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>4</mn><mo>-</mo><msup><mi>x</mi><mn>3</mn></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>ln</mi><mo> </mo><mi>x</mi></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mn>0</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>f</mi><mo>∘</mo><mi>g</mi></mrow></mfenced><mfenced><mi>x</mi></mfenced></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>f</mi><mo>∘</mo><mi>g</mi></mrow></mfenced><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>x</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence or otherwise, given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mrow><mn>2</mn><mi>a</mi></mrow></mfenced><mo>=</mo><msup><mi>f</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mn>2</mn><mi>a</mi></mrow></mfenced></math>, find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the curve <em>y</em> = 2<em>x</em><sup>3</sup> − 9<em>x</em><sup>2</sup> + 12<em>x</em> + 2, for −1 < <em>x</em> < 3</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the curve for −1 < <em>x</em> < 3 and −2 < <em>y</em> < 12.</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>A teacher asks her students to make some observations about the curve.</p>
<p>Three students responded.<br><strong>Nadia</strong> said <em>“The x-intercept of the curve is between −1 and zero”.</em><br><strong>Rick</strong> said <em>“The curve is decreasing when x < 1 ”.</em><br><strong>Paula</strong> said <em>“The gradient of the curve is less than zero between x = 1 and x = 2 ”.</em></p>
<p>State the name of the student who made an <strong>incorrect</strong> observation.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{dy}}}}{{{\text{dx}}}}"> <mfrac> <mrow> <mrow> <mtext>dy</mtext> </mrow> </mrow> <mrow> <mrow> <mtext>dx</mtext> </mrow> </mrow> </mfrac> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the stationary points of the curve are at <em>x</em> = 1 and <em>x</em> = 2.</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>Given that <em>y</em> = 2<em>x</em><sup>3</sup> − 9<em>x</em><sup>2</sup> + 12<em>x</em> + 2 = <em>k</em> has <strong>three</strong> solutions, find the possible values of <em>k</em>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p>Tommaso plans to compete in a regional bicycle race after he graduates, however he needs to buy a racing bicycle. He finds a bicycle that costs 1100 euro (EUR). Tommaso has 950 EUR and invests this money in an account that pays 5 % interest per year, <strong>compounded monthly</strong>.</p>
</div>
<div class="specification">
<p>The cost of the bicycle, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C">
<mi>C</mi>
</math></span>, can be modelled by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C = 20x + 1100">
<mi>C</mi>
<mo>=</mo>
<mn>20</mn>
<mi>x</mi>
<mo>+</mo>
<mn>1100</mn>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> is the number of years since Tommaso invested his money.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the amount that he will have in his account after 3 years. Give your answer correct to two decimal places.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the difference between the cost of the bicycle and the amount of money in Tommaso’s account after 3 years. Give your answer correct to two decimal places.</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>After <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m">
<mi>m</mi>
</math></span> complete <strong>months</strong> Tommaso will, for the first time, have enough money in his account to buy the bicycle.</p>
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m">
<mi>m</mi>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The following table shows the probability distribution of a discrete random variable <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a \geqslant 0">
<mi>a</mi>
<mo>⩾<!-- ⩾ --></mo>
<mn>0</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b \geqslant 0">
<mi>b</mi>
<mo>⩾<!-- ⩾ --></mo>
<mn>0</mn>
</math></span>.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b = 0.3 - a"> <mi>b</mi> <mo>=</mo> <mn>0.3</mn> <mo>−</mo> <mi>a</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 the difference between the greatest possible expected value and the least possible expected value.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = 6 - \ln ({x^2} + 2)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>6</mn>
<mo>−<!-- − --></mo>
<mi>ln</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</math></span>, for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \in \mathbb{R}">
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>. The graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> passes through the point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(p,{\text{ }}4)">
<mo stretchy="false">(</mo>
<mi>p</mi>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>4</mn>
<mo stretchy="false">)</mo>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p > 0">
<mi>p</mi>
<mo>></mo>
<mn>0</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span>.</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 following diagram shows part of the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span>.</p>
<p><img src="images/Schermafbeelding_2018-02-12_om_13.30.18.png" alt="N17/5/MATME/SP2/ENG/TZ0/05.b"></p>
<p>The region enclosed by the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span>, the <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 = - p"> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mi>p</mi> </math></span> and <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> is rotated 360° 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 the solid formed.</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\left( x \right) = \frac{{27}}{{{x^2}}} - 16x,\,\,\,x \ne 0">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>27</mn>
</mrow>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mo>−<!-- − --></mo>
<mn>16</mn>
<mi>x</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>≠<!-- ≠ --></mo>
<mn>0</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <em>y</em> = <em>f </em>(<em>x</em>), for −4 ≤ <em>x</em> ≤ 3 and −50 ≤ <em>y</em> ≤ 100.</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>Use your graphic display calculator to find the equation of the tangent to the graph of <em>y</em> = <em>f </em>(<em>x</em>) at the point (–2, 38.75).</p>
<p>Give your answer in the form <em>y</em> = <em>mx</em> + <em>c</em>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of the function <em>g </em>(<em>x</em>) = 10<em>x</em> + 40 on the same axes.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider a function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span>, such that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = 5, 8\,{\text{sin}}\left( {\frac{\pi }{6}\left( {x + 1} \right)} \right) + b"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>5</mn><mo>.</mo><mn>8</mn><mspace width="thinmathspace"></mspace><mtext>sin</mtext><mrow><mo>(</mo><mfrac><mi>π</mi><mn>6</mn></mfrac><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mo>)</mo></mrow><mo>+</mo><mi>b</mi></math></span>, 0 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> ≤ 10, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b \in \mathbb{R}"> <mi>b</mi> <mo>∈<!-- ∈ --></mo> <mrow> <mi mathvariant="double-struck">R</mi> </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> has a local maximum at the point (2, 21.8) , and a local minimum at (8, 10.2).</p>
</div>
<div class="specification">
<p>A second function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g">
<mi>g</mi>
</math></span> is given by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g(x) = p\,{\text{sin}}\left( {\frac{{2\pi }}{9}\left( {x - 3.75} \right)} \right) + q">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>p</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>2</mn>
<mi>π<!-- π --></mi>
</mrow>
<mn>9</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>3.75</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>q</mi>
</math></span>, 0 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> ≤ 10; <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 function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g">
<mi>g</mi>
</math></span> passes through the points (3, 2.5) and (6, 15.1).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the period of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span>(6).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span> and the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q">
<mi>q</mi>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> for which the functions have the greatest difference.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</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>4</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>x</mi><mo>+</mo><mn>4</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>4</mn></math>.</p>
</div>
<div class="specification">
<p>For the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math></p>
</div>
<div class="specification">
<p>The graphs of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> intersect at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>p</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>q</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo><</mo><mi>q</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>write down the equation of the vertical asymptote.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>find the equation of the horizontal asymptote.</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 <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></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using an algebraic approach, show that the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> is obtained by a reflection of the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> in the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis followed by a reflection in the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>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">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, find the area enclosed by the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> and the graph 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">[3]</div>
<div class="question_part_label">c.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>The following diagram shows a semicircle with centre <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math> and radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math>. Points <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P, Q</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>R</mtext></math> lie on the circumference of the circle, such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>PQ</mtext><mo>=</mo><mn>2</mn><mi>r</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>R</mtext><mover><mtext>O</mtext><mo>^</mo></mover><mtext>Q</mtext><mo>=</mo><mi>θ</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo><</mo><mi>θ</mi><mo><</mo><mi mathvariant="normal">π</mi></math>.</p>
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6qsrJCtAvSh3Ifi5BJ4yMNfQm88PMxgrpsOxBQ3Ljye9PR013ym9ui0ogniBOBDlZcHVL9+fdVLL5fSh7IE9IHEPj3tQ0WzApR8HoqTCyAthogJwF/Cw8wG9ObTORlzjIvwkJapqdnuvHMf+FCVlVvoQ1kG9unBhzp//oJ0lcC9StyB4hRlUPiAtBhSUajGs6VrMnwxrChRAKHHpJsEfKeTJ08alY6hD2UniMTnz58vqWvcq9wP5Q5s/BpF8IBfvrxICh8gTJFqVhpusHpcurRQCiDG3TvOOWsWV1xxhdr9293SBHbw4MHOWTNAc95BgwapzZtfl5lD/fr2YxNSw0HK+vbbb29rHouelklJk51PSTRg5BQF4NWgCsimwgeNiQUQHaGjOVN8p/YgdYsoKjY2RpWvXSNj5onZ6P1QWJTh3sWGXVbyRQ+KU4TRFXmoAkJ6x4bCBw1+dt0BwqQCiM6A77R9e7XzzjzgQwUCr6oJEyaqd7a9rcrKyuhDWQDacukNu/BdKVDRgeIUQbQw6Yo8lBnbRGbmw+LjpLVEezZEejcMvEEdPHjQ6IcH/h3RQR2brNEle8WKF+hDWQD28+lKPggUK/kiD8UpQuDiTUqa1CZMtlTkaWACSwfniUkqbnCcc9ZsbrrpJjlGs0t5T4E4FReXSOSE/VD79vJhZzqo5INAHT9+nAIVBShOEQAXLSKmjz5qlgeQbcKEAgg0xURzTHRytgXtO2FBYAMw2OlD2QUE6pGHM6XUnAIVWShOYUYLE0BFnm0VPvj58/IWSwFEcnKyc9YO4Inh566uNtd3ak97H6q4uJg+lOGg1DzzEQpUpKE4hZH2woQHj03oAgjcdDOmzzC+AKIj4m6JUzt31jnv7CDYh8L4jcLCQvpQhkOBijwUpzBhuzABXQCRmpLa1hLINgYMGCBHU0vKuwLihI4hGL+xrOh5+lCGQ4GKLBSnMOAFYUJHZtsKIDoiJta94YPhAB1DKiur1NChQ8WHwhh4Yi4UqMhBceolXhAm7ICHOGGfkE0FEB2BFNn1119v9H6nL2LQoBgplJg5M0XGwNOHMhsKVGSgOPUCLwgTfoeFC78vD/TU1NbfxXYGx8Vb/4CAyC5d+qzKy8unD2UBFKjwQ3HqIV4QJhRALFz4hLpw4aJ6cNZsKwsgOgL7nc6dO+eJB0RWVvZlPtTuXbudT4hpUKDCC8WpB+ChvmhRjjp79qy1wgQQMaGjQvK0ZLmxvIJuqmqr79SeYB9q/YbXxIdims9MLheoVA4t7AUUpxBp35LIVmGCx7RlS5VM/8TGQi+BBwT6AdpYsdcZ8KEwHwqNg+FDlZaWyrVIzEML1Mcfn22JfDP5d+ohFKcQCRYm2zo/aIILIDD904vEx8cbNXwwXKBxMK49+FBFRUWq6UST8wkxCQhUcC8+ClToUJxCAPOYIEwwqW0VJuTBvVYA0REYPoipvV5Mq+Da27y5QnyokpXF9KEMRffig0BlZ2c6Z0l3oTh1EwiTnscEk9pGsHrzYgFER+hBjl5K7QWDdHJtbZ0aNmwYfSiDgUAhdV5bWyvPENJ9KE7dAI1QtTDZNI+pPYsX/8STBRAdoX8/r4oTQLl5RUUVfSjDQepcDyxEOp10D4rTFwB/BpNgMVrdtnlMwaxcWaLWrQvITeK1AojOMH34YLho70MdPXLU+YSYAgYWoikxxAmLXfLFXHGxBec1aYfey3TddX1aRMqe0ertQfSQnp4mN8e8efOcs96nZnuNqqyqUDU1O6TazevgekXjXvRHRBsq27t9eA2kXRHdNjefaRGogLWVvtGCkVMnID2SnZ0lr0tKVlorTCgI0KPW586d65z1B7GxsXL0y2ZIPOxQbp6YmCiiHAgE6EMZBDxedPvXm3SZgu0ailMnoLoGaRKkTGxd4eDiz8x8RKrWsjKzPF0A0RF6+KCXfaf2YBGFvnyZLX/v+vo9slJn2yNzgBeaMTtD7kkKVNdQnDoAI8pRXYMRBrYNCwwGBRAHDhyQclavF0B0hl98p/bAH4UPhRQSxsDThzIHLJp0iTmeNaRjKE7t0CPKUQEFcbIVPxZAdMQNA2+QCkU/rlCxHwreRr9+fdVLL5eKB0fMAPekruDDvUo+D8UpCHgTWMnYXpmH32PJknwpgECVkJ9BE1iwY4c/H8z0ocxFV/DhXvWLLxoKFCeH4AKIZ54ptLYAAr8HctkogEhPT3fO+hftO6Gzh19p70OtWPECfShDwD16zTVfof/UARQnB+zetr0AAuAih9k6J2OOtQIbTlAEgtVpdbX/fKf2IBtQXFwikRN8KI6Bdx/co7pAIjPzYecsARSnFpDzrayskJWlzQUQEFiYrN++f3pbxECUirslTu3cWee88ze4vhFFxcbGyBj4rW9udT4hboF7FfvS6urq2EEiCN+Lk/ZnbPeZdIslmKwjRo5wzhIwYMAAOfqppLwrkBkIBF5VEyZMVO9se1uVlZXRh3IZbJhGZSnEiddpK74WJ+0zXXvttbLR1lYgsGixhPTV+AnjnbNEE9MSJQCvDB8MB0gnrVr1olSkHj58iD6UAWBKAPynnJzv039qwdfihMo87TPZ2t6mfQGE3zbadgc8iDEixI/7nb4IiBN9KDPAvZuakirtp9jB3MfihIauutO4zT4TCyC6x+C4eJbrdgJ9KHOIGxwnIzbggfu9QawvxQnRBlYmSIPZ7DMh8mMBRPfAfqdz585RoDqhvQ9VXFxMH8ol7h57tzyb8vIWe3JYZnfxpTihb97Zs2clnWdrtKE7WSQk3MECiG7Qv39/OdJ36hztQ2HSM9LdhYWF9KFcQDeIRUYEU6v9iu/ECWXjum/emDFjnLN2gdU/VlVYXSUnJztnSVegtyB8OVZCfTGY9PzKK+UyBn5Z0fP0oVwA16suL/dreyNfzXNCiJyUNElGeL/xRqVz1i6Qkpw0aYI6ffqMynwk07cNXXsCWvccO/a+amg44JwhXYH7BV3t0TwYWxT83grLDZBePXPmtKqq2uKLmWTB+Cpygs+EdB7aE9kKdpGjmgdVPRSm0Lj5GzdLqsTPefxQwMMQhRIzZ6bIGHj6UNEH6T14pX5M7/lGnODR6HSere2JUACBMB/hPqp6SGggYgZM7XUf+FBLlz5LH8ol/Jze84U4IRWGBzs8GlvHYAQXQHD8ds/QkSbFKXTa+1C7d+12PiGRBvc7nl2//OUvfBX1+0KcdDoP1Xk2wgKI8OHX4YPhAAVElZVVaujQoWr9htfU65teZ5ovSuj03uLFi50z3sfz4oRVMja0YbOtjdV5iPqysh5R589fkAuUHSB6B3ynU6dO0XfqIfChMB8K9xN8qNLSUrlGSWRB1I/NuVu2VEkDAT/geXFC1ITeebZutoURigKI5GnJLIAIA7GxsXLkZtzegSwExsDDhyoqKlJNJ5qcT0ikwOZctOFavPgnvlgQeFqc0OEXNw+EycbNtvj5sVLCisnPo9bDie6kQd+p92AM/ObNFeJDlawspg8VYZA1mTY1WSJ/PxRHeFackLYpKSlWo0ePlpvINhC6Q5zgkdw3/j7nLAkH9J3CBypfa2vr1LBhw+hDRQFU6erRGl5PTXtWnPLz86UIIi/PvnQeUk5I5yGERxt9El5uGHiDOnjwIL2SMIGsREVFFX2oKHH//ffL0et7nzwpTroIApNtbdvThJt64cIn1IULF9WDs2azACICoAks2LGjRo4kPLT3oY4eOep8QsIJFgN675OX09OeFKclS/KkCCI3d5Fzxh6wGsKqngUQkUP7Tg0NDXIk4UP7UNdf30+99HKpqtnOBUAkGPmtkdIrEgtZr+I5ccJmVTx0srPnWVcEwQKI6IBoFHvGqqvpO0UCZCtQbp6YmKgqqyqkpyF9qPCCa/j+5Ps9XRzhqcavSImhsSvYsaNWjraA8Dw9PU3MzoyMDOcsiRQYqIe5RY2NLIGOJOjMgs4mWAxgnx6zAeEF/Q6bm89IUYqNFcld4anICSsI5Ltzc+1qUYSqG2y0RZjOAojoMGDAADmypDyyYBsHfCg8QDEGnj5UeJkyeYo0M/Zi9OQZcULUZGPpOH5ujCXABZaVmcUCiCgRE9vaBJbDByMP7sfy8oDq168vfagwA/8U2Zb//M8Vnist94w4YeWA0vGcHLuKILDbG/Ny0lIfYMojiiAFglJ97neKDvShIse94+6Vvnu//OUvnTPewBPiFLzh1qb+eRDUdesCMsiNBRDRZ3BcPNsYRREsCDAfCls86uv3qBUrXuD4jTCA6AnPEDxLvBQ9eUKcCgoKrIua4HUsWZIvRjEnjLoD9jthxUmBii7woYqLSyRygg/FMfC9Z+w9Y+XopejJenHCSmHt2nKroqbgAoi5c+c6Z0m06d+/vxzpO0WfpKTJEkXFxsao8rVrpHqS9BxEpV6LnqwXJ0RNwJaoiQUQ5gCPDwsEVuy5A3yoQOBVNWHCRCnrLysrow/VC3T09Itf/EKOtmO1ONkYNWHfBwsgzCE+Pl7V1Gx33pFogxX/qlUvyoTqw4cP0YfqBTp6evXVdZ6InqwWp/LycjnaEjWhewXElAUQ5oDhg4hivZIKsRWIE32o3uMl78lacQre12RD1ATTfdGiXCmAGD9hvHOWuE1MTOt+J6b23Ic+VO/xkvdkrTjZtK8JQpqWliL+Rnp6On0mg6DvZBbtfSi056EPFRqjRo2S469//Ws52oq14mRT1ARhQupoTsYcWdkQs0D0xM245qB9qLy8fGlHVlhYSB8qBLDgQtcILOCxMLYVK8UJ3g2iJhvaFOXm5qj9+/erb98/vW1UAzEL+E7o7kzfySyysrLVK6+Uyxj4ZUXP04cKAd01wuaee1aKE8rH4d2YLk7BBRAjRo5wzhLTiI2NlSNTe+aBzEhlZZUaOnSo+FAYA0++GCyE0Z5r9eoy54x9WCdOeIAg1M/OznbOmAkLIOxBR7TsFGEmgwbFSKHEzJkpMgaePlT3GH/fBPXBBx/IItlGrBOnwsJnZMqtyVET8ry6AwQLIOwAOXr6TuYCH2rp0mfpQ4UAtqvgGbR69WrnjF1YJU7wBGpra6Xk1OTCgszMh9XJkydVWmoaCyAsAb4TxuPbbCD7gfY+1O5du51PSEeMu2ec2rXrPSuzAlaJU0lJq7ln8jBBdICoq6tTkyYmqbjBcc5ZYjqXhg9y1pDpaB9q2LBhav2G18SHYpqvY25PuF2OmPdkG1aJE3KnKB9HDtpE8PNhJHVCwh3qrrvvcs4SG9C+E4si7ADPgIqKKpWamiY+VGlpKaPeDkDmBs+jysoK6/59rBEn08vHETbn5S2WAojk5GTnLLEF+IL429XX1ztniA0UFBTKGHj4UEVFRarpRJPzCdHcmXinlJVXVLzhnLEDq8TJ1EIIXQBx/vwFNWP6DBZAWErcLXFq58465x2xBTwTNm+uEB+qZGUxfah26LLy5cuXO2fswApx0oUQpkZNugAiNSWVncYtBsMHAVN79oG2R7W1dfShOuFbI0epI0cOW1UYYYU46UIIE/c2sQDCO+iFBYcP2gn8FfpQHWNjYYQV4lRRUaGGDx9uXCEEcrgsgPAOeLgh/cH9TnYT7EMhlXX0yFHnE/8SXBhhC8aLEwQAFxn2N5gEwuOFC78vDzMWQHiHwXHxEgkTu9E+VL9+fdVLL5eqmu3cIjB0yFApjLClY4QF4tSq9Nh4awpIFSxc+IS6cOGienDWbBZAeAjtO7GVkf3Ah6qs3KISExNVZVWFCgQCvvahdMeI9evXO2fMxorIadKkJKM6LSBiQjeB5GnJLIDwGDGxralj+k7eAM8N9OXLzMxS9fV7xIfyc9ujOxLuUG+//RsrvDijxenS3qY054z7FBYWqC1bqtS4e+7lqHUPgocZVpfbt293zhAvkJ+/RHyo5uYzMgberz7UkCFD5Pj665vkaDJGixNSetjbZEpKD1EcxAlNQu8bf59zlniN+Ph4Rk4eBD5UeXnA1z4UKorhk69fv8E5Yy7GihPCTlSWmCJMwQUQqampzlniRdAEFpOL6Tt5D/pQSt1+W4Kqrn7H+NSeseKkW20kJSXJ0U1YAOEvMLYdNDRQnLxIex9qxYoXfOVDJSQkyNH01J7B4mROSm/x4p+wAMJH4G8M34mdIrwNfKji4hKJnOBD+WUMvL6+TU/tGStOGF1ggjBhBv+6dQEZtc4CCP+A6Imbcb0PnjGIomJjY2QM/NY3tzqfeBtU7Zme2jNSnJDSQ5Ue5ra4CVbOS5bkS7fqqdOmOmeJH4DvdOrUKenrSLwNfKhA4FU1YcJE9c62t1VZWZnnfShdtaftExMxVJzc33iLh5IetT537lznLPELsbGxcmRqzx/Ah1q16kWVk5OrDh8+5HkfClV7eLZt3WpupGikOOGBgKGCbm28RaibmfmIVGxlZWaxAMKH6OGDrNjzFxAnv/hQ2DLx1lsUp26DhwF66bkZNaEA4sCBAyot9QEWQPgY7Gej7+Q/2vtQGL/hRdBr769//aux2QHjxElvfnSrhJwFEEQD3wlVmqbvByHhJ9iHwviN4uJiz/lQulXXhg1m9tozTpxg0KEAwY3xGCyAIMEMGDBAjqgcJf5D+1B5efmSzSksLPSUD4XfD886U69vAyOnWleq9LA6ZgEECUb7TiyK8DcY1/PKK+UyBn5Z0fOe8qHibolTR48eNTI7YJQ4udkVIi0tRQog5mTMYQEEEXAdYGVZX1/vnCF+BQvmysoqNXToUE/5ULfeeqscTYyejBInvUIdMya6U2Vzc3PU/v371bfvn962WiYEYGW5cyeHD5KWSHpQjBRKzJyZ4hkfSj/vTJzxZJQ4oRgC49ijWUKOsRxr15ZLAcSIkSOcs4S0oocPMrVHAJ5NS5c+e5kP1XSiyfnUTlCVWlf3rvPOHIwRJ+Q8Gxoa1OjR0fObULa+aFGupG7GTxjvnCXkEnorAUdokGCCfaiSlcVq967dzif2garUDz74wDjfyRhx0psdo1UMgT8EfCYUQKSnp9NnIh2ClTLGpHC/E2mP9qGGDRum1m94TXwoG9N8l7qhmOU7GSNOemUaLb8puAAimmlEYh+D4+JVXR19J/J54ENVVFSp1NQ08aFKS0uNrHzrCu07vfXWb+RoCsaIE1Qb6bVoCEV+fh4LIEi30b4TWxmRzigoKJQx8PChioqKrPOh8Ow1LXVtUOQUnf1NKIBYtWqlSki4gwUQpFvonfT0nUhXYAz85s0VVvpQqEp9//2jzjszMEKcouU34f+Tl7dYVgnJycnOWUK6BtE8vMnt27c7ZwjpGLQ9qq2ts86HMrEq1Qhx0uOwhw+PXC875IHRAeL8+QtqxvQZLIAgIYEOzoycSHfAYqa9D2V62yNdlfq73/1WjiZgVOSEVUekyMx8WJ08eVKlpqSy0zgJGZTbooCGvhPpLtqHam4+I+M3jh4xK20WjM4O1NSYU7FnTOSE+U2RAgUQqLaaNDFJhmwREioY2w50lE9Id4APVV4eUP369VUvvVyqarab20QY1/iBA/udd+5jhDihGCJSKb3gAoi77o5uWyTiHRBtY2XpVk5++fIiFRs7SL6mTZvashpvdj4hpoOMUGXlFpWYmKgqqypUIBAw0ofSm3FNwXVximRKjwUQJJxgZenGZtz58x9T27ZtU42NTWrTptdbjsfV6tVlzqfEBpA2Q1++zMwsVV+/x0gf6tKIGDO8VdfFSadJwj2/iQUQJNxgZXnq1CnV1HTCORN5EDHhYbZ69X/Le2QAxo69R23cuFHeE7vIz19irA+l95jiejMB18Wpqal1s1q4y8gXLvw+CyBIWLnU5iU6K0uk7oqKlqmMjDnOmUvg4UbsxFQfSj8n33//mBzdxnVx0p0hwklhYYHasqVKjbvnXhZAkLChu4lEq2IPqTsI1OzZGc6ZVhobG51XxFba+1BlZWVG+FDoUL53rxnzy1wXpxMnmlRMGFsIYWAhxAn/yPeNv885S0h4wHUVLd9pz57W9EpCwm1txRD4QtoF6T1iN9qHysnJVYcPH1IrVrzgug/Vr28/dfToEeedu7guTuhFFa5mr1jRIp2HLtKpqanOWULCB3yngwcPRqW5Z3X1NjV//gIphNBfy5e/IJ/pFCOxH4hTcXGJRE7wodwcA9+3b1917tw5I5rXuipOOj0yaFDvIyf8Yy5c+IS6cOGienDWbBZAkIhwqaIpsj4BUndI6eFhEQyq9gCrT71FUtJkiaJiY2NkDPzWN7c6n0QXveiJVuq6K1wVJ131FI5KPURMWNEmT0tmAQSJGNp3inRRBMrFQZ8+feQIIFabNm2Uaj2m9bwHfKhA4FU1YcJE9c62t13xofSiPpoVqZ3hqjhh8i3o7R6n4AKI226PXAskQnDzooCnvj6ypnGwKGl0gcSTTz7pnCFeAz7UqlUvuuZD6YX9oUOH5OgmrooTbjSg6+t7AlawLIAg0QTjBXbujOzwQURG+NL7mSBMTz31M/X00z9n1OQD3PSh4NkjC+U2V1xswXkddR54oLVoYc2atXIMFYSekyZNVJ99dr7lj5lDn4lEBWycxP4UFCtcffXVssgK7rmHNDW6ScBLRVuunmYGUJWH7hDwn+AFLFjw+OfKyom3aS3yekLEInHUnWrqtKnOJ5ED6cRrrrmmZWG0yTnjDq6K0+TJkyRq6ok4oQAiNXWmOnDggHp8wffoM5GogWuvoOAZ510rwY2L0SsymGuvvVYM76SkJDkSEgq43hYv/olaty4gKeW5c+dGdCGOGVT7Gvaq/fvdjZ5cFSfs2UD4iq9QQQEE/lhpqQ/QZyJR57nnnpXKPczt6QyknBFR4YjqvrNnz4pQZWfPU1lZ2b1KZxP/sXJliVqyJF8aEGdlZkVsQY6OFdgYjK0LbuKa59SbOnr8kSBMCHMpTMQNBsfFt6wsux4vgJZcEKGSkpUt4vSu9FRDug8e6Zgxd8p1TEh3wbX0yivlMgZ+WdHzEfOhkKoGbpeTuyZO+hcfPny4HLsLVqFYPSC8jUb+lZCO0GOtu3sDI0pCT7U33qiUBwxECtcxUtsmlO0SO8CCp7KySg0dOlT2QyEFF2769+8vx94EEOHA1Wo9cN11ny+Z7QzcxOg0jrAWeVdC3CImtnVvXk9Gt+MBA5FCOhvbKZKSJknbLUK6AwpusGF35swUGQNfXFwckf1QH33k7sww18RJVzd1N+8OFc/MfERGZSPfyso84ia4brFI2r59u3MmdCBOmzdXyAJt3rxsGYxJSHfA9bd06bMqLy9fWsAVFhaqphPh8Yj0RnO9D9UtXPecultmi1HrqMxDAQQr84gJxMfH9yhyCgbXf0VFpaS3Fy3KpUCRkAj2oUpWFqvdu3Y7n/Sev/3tU+eVO7ie1usOuGHXri1nAQQxCjSBRSTfW+MYq2Bsp6BAkZ6ANHFtbZ0aNmyYWr/hNfGhepvmQ1YAEyPcxNWCiO7MccL34YbF946fMN45S4j7oKgBBG/A7SnBAoUsQW8Fj/gLXD/Y1pCamiY+VGlpaa8KGnBtY1irm7iY1mtu+QfoWpzwj5uWliIqnp6eTp+JGAXSy7g2w9UEFg+YZ54plNeLFuXIkZBQKCgolC0L8KGKiorC5kO5gdFpPQgT0iZzMubIjUuIacB3CufwQXhQubmLxIzGfihCQgVbFlBo01sf6h//+Ifzyh2MFafc3BzZ5Pjt+6e3VY8QYho3DLxBnTp1Kqx7lWByox1SSUkx90CRHoFFTm98KFzXe/b83nnnDq6JE/qPoSlmRwQXQIwYOcI5S4h56OFs4UrtaXJyFkm7o4ICRk+kZ3TkQ3V3/IbuEuEmrkZOHc2sYQEEsYlIDR9EBRaiJyzS3N6pT+xG+1DNzWdk/Aa66tuAUWk93IS6AwQLIIgtYJYYxluEG0RPgKXlpLfAhyovD6h+/frKuBc0dzUdo8QpM/NhKV9MawlDWQBBbAH7nTBvJ9wRDqInZBAoTiQcwIeqrNyiEhMTpet4IBCISNujcGGMOGFvR11dnZo0MUnFDY5zzhJiPpd8p/CvRjEDCpV7TO2RcIBFP/ryZWZmSbTfmQ+FcTAg3OnqUDBCnLAyXLVqpYyfvuvuu5yzhNiBbqcViRsZ0ROIhPAR/5Kfv6RLH8oES8UVcdI3MXbDowAiL2+xpC+Sk5PlPCE2gRsZ129NTc+bwHbGmDGtizW3m3AS72G6D+Vq5HTllVdKAcT58xfUjOkzWABBrCXuljhpTBxukIaB8LGdEYkE7X2osrIyY3woV8XpP/7j51IAkZqSyk7jxGr08MFIpPbQ5svt2TrEu2gfCiNcDh8+pFaseEGdPn3a+dQ9XBUndIBgAQTxAnq/U29HaBDiFhCn4uISiZw2btrgnHWPKy624LyOGlhdpqenOe8IIYSYCGZF6aKcaOOKOKFf2EsvvSSvTWiTQQgh5BKffto6aPChhx6SsfBu4Io4EUIIIV1hxD4nQgghJBiKEyGEEOOgOBFCCDEOihMhhBDj6JY4TZs2VcXGDvrc1+zZD6pNmzY630WIv8C1P3/+Y5fdEz/+8Y9UY2Oj8x2E+ANc87j2g++F3upDt8Rp06bX1dNP/7ztdWNjkxybm5vl5qyu3iafEeIXnnrqZ44wxcr9gK/q6hq5SadNm9Krm5IQm8C1jmse137rPdB6P+DewD2yfHmR850hglLy7rBx44aLMTFfv7hnz++dM5fOPfjgLOcMId6nrOxlue5/9rP/45y5nKlTp1y8/fbhF48fP+6cIcSb4BrHtY5rviN+9KMfyr0CrQiVXnlOY8feI0dEUIT4haKiZXLMyJgjx/ZkZGTIPaG/jxCvgmsc1zqu+Y5YsOBxORYVhR499UqcGhuPy7FPnz5yJMTrYEAb0hdIWeghg+1JSEiQI9PdxOvoa1wHKu3R94m+b0IhZHGqrq6WY6sB9mN53ZlqEuJV+vTp67zqHGYUiNfR13hXAYq+VzDYMBRCFicYwajEGDv2LvmfoVBi2jQOCST+ojs3GjMKxC90tRDT90p3FnTBhCxOulpPVyfNns2oifiHhIQ7JE2BzEFnaYr6+no5ctFGvI5O53WWwoZo4aurNHhn9MpzIsSP6EKIzgoeME0UUZM2gwnxKtrSwTXfEatXl4k49eRe6LY46VViV+EbIX5g/vwF8oUbDxsPNbhHsPEQhUKrV/8303rE8yByevLJf5eCh9Zr/1I2AfdG637ABT3KsHVrZAY6ROB/rkG6YvnyF5x3hPgTbD7EijE4pYEbEatEChPxEx3dC0iBI7LqqfXDeU6EEELCQnA2AREVFms9heJECCEkbCC1B4FCFIW0H6KnnhQHUZwIIYSEHVhBGzdulJQfBKu+fl9I6W6KEyGEEONgKTkhhBDDUOr/A1Q1Zm59oDg0AAAAAElFTkSuQmCC"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that the areas of the two shaded regions are equal, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mo>=</mo><mn>2</mn><mo> </mo><mi>sin</mi><mo> </mo><mi>θ</mi></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence determine the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</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>
<br><hr><br><div class="specification">
<p>The following table shows a probability distribution for the random variable <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{E}}(X) = 1.2">
<mrow>
<mtext>E</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>1.2</mn>
</math></span>.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-15_om_06.18.09.png" alt="M17/5/MATME/SP2/ENG/TZ2/10"></p>
</div>
<div class="specification">
<p>A bag contains white and blue marbles, with at least three of each colour. Three marbles are drawn from the bag, without replacement. The number of blue marbles drawn is given by the random variable <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span>.</p>
</div>
<div class="specification">
<p>A game is played in which three marbles are drawn from the bag of ten marbles, without replacement. A player wins a prize if three white marbles are drawn.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q"> <mi>q</mi> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the probability of drawing three blue marbles.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain why the probability of drawing three white marbles is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{6}"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The bag contains a total of ten marbles of which <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w"> <mi>w</mi> </math></span> are white. Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w"> <mi>w</mi> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Jill plays the game nine times. Find the probability that she wins exactly two prizes.</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>Grant plays the game until he wins two prizes. Find the probability that he wins his second prize on his eighth attempt.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = - {x^4} + a{x^2} + 5">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>a</mi>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>5</mn>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> is a constant. Part of the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f(x)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span> is shown below.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-16_om_17.47.40.png" alt="M17/5/MATSD/SP2/ENG/TZ2/06"></p>
</div>
<div class="specification">
<p>It is known that at the point where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 2">
<mi>x</mi>
<mo>=</mo>
<mn>2</mn>
</math></span> the tangent to the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f(x)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span> is horizontal.</p>
</div>
<div class="specification">
<p>There are two other points on the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f(x)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span> at which the tangent is horizontal.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-intercept of the graph.</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'(x)">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</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="a = 8">
<mi>a</mi>
<mo>=</mo>
<mn>8</mn>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(2)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-coordinates of these two points;</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the intervals where the gradient of the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f(x)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span> is positive.</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>Write down the range of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the number of possible solutions to the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = 5">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>5</mn>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = m">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>m</mi>
</math></span>, where <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>, has four solutions. Find the possible values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m">
<mi>m</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.</div>
</div>
<br><hr><br><div class="specification">
<p>All living plants contain an isotope of carbon called carbon-14. When a plant dies, the isotope decays so that the amount of carbon-14 present in the remains of the plant decreases. The time since the death of a plant can be determined by measuring the amount of carbon-14 still present in the remains.</p>
<p>The amount, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math>, of carbon-14 present in a plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> years after its death can be modelled by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><msub><mi>A</mi><mn>0</mn></msub><msup><mtext>e</mtext><mrow><mo>-</mo><mi>k</mi><mi>t</mi></mrow></msup></math> where <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"><msub><mi>A</mi><mn>0</mn></msub><mo>,</mo><mo> </mo><mi>k</mi></math> are positive constants.</p>
<p>At the time of death, a plant is defined to have <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn></math> units of carbon-14.</p>
</div>
<div class="specification">
<p>The time taken for half the original amount of carbon-14 to decay is known to be <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5730</mn></math> years.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>A</mi><mn>0</mn></msub><mo>=</mo><mn>100</mn></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"><mi>k</mi><mo>=</mo><mfrac><mrow><mi>ln</mi><mo> </mo><mn>2</mn></mrow><mn>5730</mn></mfrac></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find, correct to the nearest <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> years, the time taken after the plant’s death for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>25</mn><mo>%</mo></math> of the carbon-14 to decay.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi><mo>+</mo><mfrac><mn>50</mn><mi>x</mi></mfrac><mo>,</mo><mo> </mo><mo> </mo><mi>x</mi><mo>≠</mo><mn>0</mn><mo>.</mo></math></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mn>1</mn></mfenced></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>0</mn></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>The graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> has a local minimum at point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>.</p>
<p>Find the coordinates of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</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{{48}}{x} + k{x^2} - 58">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>48</mn>
</mrow>
<mi>x</mi>
</mfrac>
<mo>+</mo>
<mi>k</mi>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>58</mn>
</math></span>, where <em>x</em> > 0 and <em>k</em> is a constant.</p>
<p>The graph of the function passes through the point with coordinates (4 , 2).</p>
</div>
<div class="specification">
<p>P is the minimum point of the graph of <em>f </em>(<em>x</em>).</p>
</div>
<div class="question">
<p>Sketch the graph of <em>y</em> = <em>f</em> (<em>x</em>) for 0 < <em>x</em> ≤ 6 and −30 ≤ <em>y</em> ≤ 60.<br>Clearly indicate the minimum point P and the <em>x</em>-intercepts on your graph.</p>
</div>
<br><hr><br><div class="specification">
<p>A water container is made in the shape of a cylinder with internal height <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h">
<mi>h</mi>
</math></span> cm and internal base radius <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span> cm.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-03-07_om_08.31.01.png" alt="N16/5/MATSD/SP2/ENG/TZ0/06"></p>
<p>The water container has no top. The inner surfaces of the container are to be coated with a water-resistant material.</p>
</div>
<div class="specification">
<p>The volume of the water container is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0.5{\text{ }}{{\text{m}}^3}">
<mn>0.5</mn>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>m</mtext>
</mrow>
<mn>3</mn>
</msup>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>The water container is designed so that the area to be coated is minimized.</p>
</div>
<div class="specification">
<p>One can of water-resistant material coats a surface area of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2000{\text{ c}}{{\text{m}}^2}">
<mn>2000</mn>
<mrow>
<mtext> c</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>m</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down a formula for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A"> <mi>A</mi> </math></span>, the surface area to be coated.</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>Express this volume in <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{c}}{{\text{m}}^3}"> <mrow> <mtext>c</mtext> </mrow> <mrow> <msup> <mrow> <mtext>m</mtext> </mrow> <mn>3</mn> </msup> </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>Write down, in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r"> <mi>r</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h"> <mi>h</mi> </math></span>, an equation for the volume of this water container.</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>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = \pi {r^2} + \frac{{1\,000\,000}}{r}"> <mi>A</mi> <mo>=</mo> <mi>π</mi> <mrow> <msup> <mi>r</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mfrac> <mrow> <mn>1</mn> <mspace width="thinmathspace"></mspace> <mn>000</mn> <mspace width="thinmathspace"></mspace> <mn>000</mn> </mrow> <mi>r</mi> </mfrac> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}A}}{{{\text{d}}r}}"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>A</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>r</mi> </mrow> </mfrac> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using your answer to part (e), find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r"> <mi>r</mi> </math></span> which minimizes <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A"> <mi>A</mi> </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>Find the value of this minimum area.</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the least number of cans of water-resistant material that will coat the area in part (g).</p>
<div class="marks">[3]</div>
<div class="question_part_label">h.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the lines <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_1}">
<mrow>
<msub>
<mi>L</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_2}">
<mrow>
<msub>
<mi>L</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span> with respective equations</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_1}\,{\text{:}}\,y = - \frac{2}{3}x + 9">
<mrow>
<msub>
<mi>L</mi>
<mn>1</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>:</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
<mi>x</mi>
<mo>+</mo>
<mn>9</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_2}{\text{:}}\,y = \frac{2}{5}x - \frac{{19}}{5}">
<mrow>
<msub>
<mi>L</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow>
<mtext>:</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
<mo>=</mo>
<mfrac>
<mn>2</mn>
<mn>5</mn>
</mfrac>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mfrac>
<mrow>
<mn>19</mn>
</mrow>
<mn>5</mn>
</mfrac>
</math></span>.</p>
</div>
<div class="specification">
<p>A third line, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_3}">
<mrow>
<msub>
<mi>L</mi>
<mn>3</mn>
</msub>
</mrow>
</math></span>, has gradient <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \frac{3}{4}">
<mo>−<!-- − --></mo>
<mfrac>
<mn>3</mn>
<mn>4</mn>
</mfrac>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the point of intersection of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_1}"> <mrow> <msub> <mi>L</mi> <mn>1</mn> </msub> </mrow> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_2}"> <mrow> <msub> <mi>L</mi> <mn>2</mn> </msub> </mrow> </math></span>.</p>
<div 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 a direction vector for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_3}"> <mrow> <msub> <mi>L</mi> <mn>3</mn> </msub> </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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_3}"> <mrow> <msub> <mi>L</mi> <mn>3</mn> </msub> </mrow> </math></span> passes through the intersection of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_1}"> <mrow> <msub> <mi>L</mi> <mn>1</mn> </msub> </mrow> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_2}"> <mrow> <msub> <mi>L</mi> <mn>2</mn> </msub> </mrow> </math></span>.</p>
<p>Write down a vector equation for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_3}"> <mrow> <msub> <mi>L</mi> <mn>3</mn> </msub> </mrow> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>3</mn><mi>x</mi><mo>-</mo><msup><mn>4</mn><mrow><mn>0</mn><mo>.</mo><mn>15</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></msup></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>3</mn></math>.</p>
</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> on the grid below.</p>
<p><img src="data:image/png;base64,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"></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> for which <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>′</mo><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>0</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>The quadratic equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mo>+</mo><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>3</mn></mrow></mfenced><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>, has real distinct roots.</p>
<p>Find the range of possible values for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>.</p>
</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"><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">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The functions <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> are defined for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math> by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>12</mn><mi>x</mi><mo>+</mo><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>-</mo><mi>x</mi><mo>+</mo><mi>c</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the range of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</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"><mfenced><mrow><mi>g</mi><mo>∘</mo><mi>f</mi></mrow></mfenced><mfenced><mi>x</mi></mfenced><mo>≤</mo><mn>0</mn></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>, determine the set of possible values for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</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><msup><mi mathvariant="normal">e</mi><mrow><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></msup><mo>-</mo><mn>0</mn><mo>.</mo><mn>5</mn></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>2</mn></math>.</p>
</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>x</mi></math> for which <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>0</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> on the following grid.</p>
<p><img src="data:image/png;base64,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"></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>SpeedWay airline flies from city <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}">
<mrow>
<mtext>A</mtext>
</mrow>
</math></span> to city <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{B}}">
<mrow>
<mtext>B</mtext>
</mrow>
</math></span>. The flight time is normally distributed with a mean of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="260">
<mn>260</mn>
</math></span> minutes and a standard deviation of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="15">
<mn>15</mn>
</math></span> minutes.</p>
<p>A flight is considered late if it takes longer than <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="275">
<mn>275</mn>
</math></span> minutes.</p>
</div>
<div class="specification">
<p>The flight is considered to be <strong>on time</strong> if it takes between <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m">
<mi>m</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="275">
<mn>275</mn>
</math></span> minutes. The probability that a flight is on time is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0.830">
<mn>0.830</mn>
</math></span>.</p>
</div>
<div class="specification">
<p>During a week, SpeedWay has <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="12">
<mn>12</mn>
</math></span> flights from city <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}">
<mrow>
<mtext>A</mtext>
</mrow>
</math></span> to city <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{B}}">
<mrow>
<mtext>B</mtext>
</mrow>
</math></span>. The time taken for any flight is independent of the time taken by any other flight.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the probability a flight is <strong>not</strong> late.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m"> <mi>m</mi> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the probability that at least <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="7"> <mn>7</mn> </math></span> of these flights are <strong>on time</strong>.</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>Given that at least <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="7"> <mn>7</mn> </math></span> of these flights are on time, find the probability that exactly <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="10"> <mn>10</mn> </math></span> flights are on time.</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>SpeedWay increases the number of flights from city <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}"> <mrow> <mtext>A</mtext> </mrow> </math></span> to city <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{B}}"> <mrow> <mtext>B</mtext> </mrow> </math></span> to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="20"> <mn>20</mn> </math></span> flights each week, and improves their efficiency so that more flights are on time. The probability that at least <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="19"> <mn>19</mn> </math></span> flights are on time is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0.788"> <mn>0.788</mn> </math></span>.</p>
<p>A flight is chosen at random. Calculate the probability that it is on time.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>A particle moves in a straight line such that its velocity, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo> </mo><mtext>m s</mtext><msup><mrow></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>, at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> seconds is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>=</mo><mfrac><mrow><mfenced><mrow><msup><mi>t</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced><mi>cos</mi><mo> </mo><mi>t</mi></mrow><mn>4</mn></mfrac><mo>,</mo><mo> </mo><mn>0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mn>3</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine when the particle changes its direction of motion.</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 times when the particle’s acceleration is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1</mn><mo>.</mo><mn>9</mn><mo> </mo><msup><mtext>m s</mtext><mrow><mo>-</mo><mn>2</mn></mrow></msup></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 particle’s acceleration when its speed is at its greatest.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</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) = {x^3} - 5{x^2} + 6x - 3 + \frac{1}{x}">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>5</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>6</mn>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>3</mn>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mi>x</mi>
</mfrac>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x > 0">
<mi>x</mi>
<mo>></mo>
<mn>0</mn>
</math></span></p>
</div>
<div class="specification">
<p>The function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = {x^3} - 5{x^2} + 6x - 3 + \frac{1}{x}">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>5</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>6</mn>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>3</mn>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mi>x</mi>
</mfrac>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x > 0">
<mi>x</mi>
<mo>></mo>
<mn>0</mn>
</math></span>, models the path of a river, as shown on the following map, where both axes represent distance and are measured in kilometres. On the same map, the location of a highway is defined by the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( x \right) = 0.5{\left( 3 \right)^{ - x}} + 1">
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0.5</mn>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mn>3</mn>
<mo>)</mo>
</mrow>
<mrow>
<mo>−<!-- − --></mo>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</math></span>.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p>The origin, O(0, 0) , is the location of the centre of a town called Orangeton.</p>
<p>A straight footpath, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P">
<mi>P</mi>
</math></span>, is built to connect the centre of Orangeton to the river at the point where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{1}{2}">
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math></span>.</p>
</div>
<div class="specification">
<p>Bridges are located where the highway crosses the river.</p>
</div>
<div class="specification">
<p>A straight road is built from the centre of Orangeton, due north, to connect the town to the highway.</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="f\left( x \right)">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{1}{2}">
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math></span>.</p>
<div 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 function, <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> , that would define this footpath on the map.</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 of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P">
<mi>P</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the coordinates of the bridges relative to the centre of Orangeton.</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 distance from the centre of Orangeton to the point at which the road meets the highway.</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>This straight road crosses the highway and then carries on due north.</p>
<p>State whether the straight road will ever cross the river. Justify your answer.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = {{\text{e}}^{2\,{\text{sin}}\left( {\frac{{\pi x}}{2}} \right)}}">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mi>π<!-- π --></mi>
<mi>x</mi>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</msup>
</mrow>
</math></span>, for <em>x</em> > 0.</p>
<p>The <em>k </em>th maximum point on the graph of <em>f</em> has <em>x</em>-coordinate <em>x<sub>k</sub></em> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k \in {\mathbb{Z}^ + }">
<mi>k</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <em>x<sub>k</sub></em><sub> + 1</sub> = <em>x<sub>k</sub></em> + <em>a</em>, find <em>a</em>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the value of <em>n</em> such that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sum\limits_{k = 1}^n {{x_k} = 861} "> <munderover> <mo movablelimits="false">∑</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <mrow> <mrow> <msub> <mi>x</mi> <mi>k</mi> </msub> </mrow> <mo>=</mo> <mn>861</mn> </mrow> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>OAB is a sector of the circle with centre O and radius <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span>, as shown in the following diagram.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p>The angle AOB is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta ">
<mi>θ<!-- θ --></mi>
</math></span> radians, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 < \theta < \frac{\pi }{2}">
<mn>0</mn>
<mo><</mo>
<mi>θ<!-- θ --></mi>
<mo><</mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>2</mn>
</mfrac>
</math></span>.</p>
<p>The point C lies on OA and OA is perpendicular to BC.</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="{\text{OC}} = r\,{\text{cos}}\,\theta "> <mrow> <mtext>OC</mtext> </mrow> <mo>=</mo> <mi>r</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</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 the area of triangle OBC in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r"> <mi>r</mi> </math></span> and <em>θ</em>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that the area of triangle OBC is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{3}{5}"> <mfrac> <mn>3</mn> <mn>5</mn> </mfrac> </math></span> of the area of sector OAB, find <em>θ</em>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = x{{\text{e}}^{ - x}}">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>x</mi>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−<!-- − --></mo>
<mi>x</mi>
</mrow>
</msup>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g(x) = - 3f(x) + 1">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mn>3</mn>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mn>1</mn>
</math></span>.</p>
<p>The graphs of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g">
<mi>g</mi>
</math></span> intersect 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="x = q">
<mi>x</mi>
<mo>=</mo>
<mi>q</mi>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p < q">
<mi>p</mi>
<mo><</mo>
<mi>q</mi>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span> and of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q">
<mi>q</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>Hence, find the area of the region enclosed by the graphs of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g">
<mi>g</mi>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = x - 8">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>8</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( x \right) = {x^4} - 3">
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>3</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h\left( x \right) = f\left( {g\left( x \right)} \right)">
<mi>h</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h\left( x \right)"> <mi>h</mi> <mrow> <mo>(</mo> <mi>x</mi> <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>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{C}}"> <mrow> <mtext>C</mtext> </mrow> </math></span> be a point on the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h"> <mi>h</mi> </math></span>. The tangent to the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h"> <mi>h</mi> </math></span> at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{C}}"> <mrow> <mtext>C</mtext> </mrow> </math></span> is parallel to the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span>.</p>
<p>Find the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-coordinate of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{C}}"> <mrow> <mtext>C</mtext> </mrow> </math></span>.</p>
<div class="marks">[5]</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></math> defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>90</mn><msup><mtext>e</mtext><mrow><mo>-</mo><mn>0</mn><mo>.</mo><mn>5</mn><mi>x</mi></mrow></msup></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><msup><mi mathvariant="normal">ℝ</mi><mo>+</mo></msup></math>.</p>
<p>The graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> and the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi></math> intersect at point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>.</p>
</div>
<div class="specification">
<p>The line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> has a gradient of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1</mn></math> and is a tangent to the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Q</mtext></math>.</p>
</div>
<div class="specification">
<p>The shaded region <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> is enclosed by the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> and the lines <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math>.</p>
<p style="text-align: center;"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAakAAAEKCAYAAACopKobAAAgAElEQVR4Ae2d3Y8c1ZmHI5E/gVwldgyWFRibZAPrDzCRNojYgLRKgDW2L1YQjJSRNlkFKQRZIXhzgSMItkI2+AI70kbCkGBACjiLpcxygaXZhchSsKzkikiOc23Zf0GtfmVeT01PdXd913vOeY40qu7qqlPved7qfuZUnar6TEaBAAQgAAEIOCXwGadxERYEIAABCEAgQ1LsBBCAAAQg4JYAknKbGgKDAAQgAAEkxT4AAQhAAAJuCSApt6khMAhAAAIQQFLsAxCAAAQg4JYAknKbGgKDAAQgAAEkxT4AAQhAAAJuCSApt6khMAhAAAIQQFLsAxCAAAQg4JYAknKbGgKDAAQgAAEkxT4AAQhAAAJuCUQvqU8++Wt2ww2fzf/27t13PREHDx7MFhcXr7/nBQQgAAEI+CMQvaQMuQR1442fs7fZK68cz3bt2n39PS8gAAEIQMAfgWQk9fzzL+S9KUvBuXPnMs2jQAACEICAXwLJSEo9Jx320+E/FR3uu3z5st/MEBkEIAABCKTzqA6TlHpQS0tL9KLY+SEAAQgEQCCZnpTEpJ6UpsUBFAHkiBAhAAEIJEsgGUmpByVJabCEREWBAAQgAAH/BJKRlA1F17koCgQgAAEIhEEgKUlt3botjKwQJQQgAAEI5ASSkZQO89nIPnIPAQhAAAJhEIhaUhrRp96T7iyhc1Iqly79LYzMRBblmTPvwT6ynFZpzhtv/Da7cOFClUVZBgKlBKKWlF3AawMl9IVZt+7zmaaUYQns2fMw3IdFPvrW9D1bWLgFSY2eibADiFpSxdSYoCQpRFUkM8zro0ePZIcOPVtpY4cPP5fnaMuWhezixYuV1mEhXwQQlK98hBxNEpIyQem/OgnKpppPGYbA8vJypt7UvCJBPfDA/bmclKvz5z+etwqfOyOAoJwlJPBwopeUjodLSnbYQT98Ns9eB57DIMK/evVq/g/CrGDVa1JOzp79YNZifOaYAIJynJxAQ4taUiYjE5RypB9BlbLPAs1hMGHv2LE9U49qWtm/f1+2c+ed0z5mvnMCCMp5ggINL1pJaRRf2WE9k5TypRFneq/lGPXX/x785JPfz06cOL5mQ8eOvZznQbnQnw73UcIigKDCyldI0UYrKSVBJ+r15SmWoqQ0X59XPaFfrIfX9QlIUBJVWbly5UouKAmLEhYBBBVWvkKLNmpJlSVjUlJlyzCvHwI6xKpDfmXl5MlXc0lJVpRwCCCocHIVaqRIKtTMBRq3/knQIIrJsrj4HQ7zTUJx/h5BOU9QJOEhqUgSGUozNAxd5wKLRb0nXRPFob4iFd+vEZTv/MQUHZKKKZsBtEXn/3Rhb7GcPv1ufqiPC3eLVPy+RlB+cxNjZEgqxqw6bpN6UZMX9T799A8zDT+n+CeAoPznKLYIkVRsGXXeHg31nxy8okN9XMDrPHGfjoQtXnPoP2IijIFAkpIqO3EfQzJDacP27VuvH95TL0qDJii+CdCD8p2fmKNLUlK7d3+jdIRZzIn21LZHH/3X7Mtf3pKLCkF5ykx5LBKUer+6hIACgaEJJCkpXVCKqIbe1Va2N+ui3pWleOWBgAlKUwoExiCQpKQEGlGNsbtd2+asi3rHi4otTxJAUJNEeD8GgWQlJdiIaoxd7to2p13UO15EbLlIAEEVafB6TAJJS0rgEdU4u1/ZRb3jRMJWJwkgqEkivB+TQPKSEnxENfwuWOdJvcNHl+4WEVS6uffaciT1aWYQ1bC7aNlFvcNGwNYmCSCoSSK890AASRWygKgKMHp+WeVJvT2HQPUFAvZsNYmKAgFPBJDURDYQ1QSQHt/qMoBZT+rtcdNUXSBgT6lGUAUovHRDAEmVpAJRlUDpYZY4lz2pt4dNUeUUAghqChhmuyGApKakAlFNAdPhbP3nfuDA4x3WSFV1CCCoOrRYdiwCSGoGeUQ1A04HH9mPZAdVUUVNAsaeQ3w1wbH44ASQ1BzkiGoOoJYf667aujM6ZTgCCGo41mypPQEkVYEhoqoAqeEiOtzHf/MN4TVYDUE1gMYqoxJAUhXxI6qKoGoupot6xZbSPwEE1T9jttA9ASRVgymiqgGr4qIagr5jx/aKS7NYUwIIqik51hubAJKqmQFEVRNYhcW52WwFSC0WQVAt4LHq6ASQVIMUIKoG0Gasws1mZ8Bp+RGCagmQ1UcngKQapgBRNQRXsho3my2B0sEsBNUBRKoYnQCSapECRNUCXmFVbjZbgNHRSwTVEUiqGZ0AkmqZAkTVEmCWZdxstj3DYg0IqkiD16ETQFIdZBBRtYfIzWbbM1QNCKobjtTihwCS6igXiKodyEOHns10borSnACCas6ONf0SQFId5gZRNYfJeanm7LQmgmrHj7X9EkBSFXJz9uwHma7lKf6dPPlq6ZqIqhTL3Jm6f5/4UuoTQFD1mbFGOASQVMVcXblyJf8RfeCB++eugajmIipdQHee0A8upToBBFWdFUuGSQBJVczb+fMf55Ka1oOarAZRTRKZ/17MeAjifE62BIIyEkxjJoCkKmb38OHnckmpR1W1IKqqpK4tx0MQq/NCUNVZsWTYBJBUxfzpMN/+/fsqLr2yGKJaYTHvlc5L6flSlNkEENRsPnwaFwEkVSGfFy9ezHtRx469XGHptYsgqrVMps3hvNQ0MtfmI6jZfPg0PgJIqkJOJSeNPNN5qaYFUVUjp4cgcl6qnBWCKufC3LgJIKkK+dVhviqj+uZVhajmEcpyQUlUlNUEENRqHrxLhwCSmpNrG3qugRPFovmLi98pzqr0GlHNxmQ/xrOXSutTY6K7clAgkBoBJDUn4zaqr3ioT6/Vs+Ic1Rx4DT/W4An9MFNW7iShf24oEEiRAJKaknXrQRXvMjH5us5w9MnN0KOaJLLynvNS11hYDwpBrewbvEqPAJIaMeeIqhy+Bk6kfl4KQZXvG8xNjwCSGjnniGptAvQDraHoqRYElWrmaXcZASRVRmXgeYhqLXCdl9LFvakVBJVaxmnvPAJIah6hgT5HVKtB63CfbpOUUkFQKWWbtlYlgKSqkhpgOUS1Ajm181JXr17N9HRi7QMUCEBghQCSWmHh4hWiupYG61W4SErPQSCongFTfdAEkJTD9CGqa0lJ4XopBOXwC0hIrgggKVfpWAkGUWX5MPSY7+OHoFb2d15BYBoBJDWNjIP5qYsq5vNSCMrBF4wQgiCApJynKWVRxXpeCkE5/9IRnisCSMpVOsqDSVlUsT1fCkGV7+PMhcA0AkhqGhln81MVldody3kpBOXsS0U4QRBAUkGk6VqQKYpKF/TGcB8/BBXQF41QXRFAUq7SMT+Y1ESlWyNpKHrIBUGFnD1iH5sAkho7Aw22n5qoQj4vhaAa7OCsAoECASRVgBHSy5REpbYePXokpPTksSKo4FJGwA4JICmHSakaUiqiOnPmvWzPnoerYnGxHIJykQaCiIAAkgo8iSmISj/4eipyKAVBhZIp4gyBAJIKIUtzYkxBVLpDuHpU3guC8p4h4guNAJIKLWNT4o1dVIcOPZvpz3NBUJ6zQ2yhEkBSoWauJO6YRaVelHpTXguC8poZ4gqdAJIKPYMT8ccsKp2Xkgy8FQTlLSPEExMBJBVTNj9tS6yi0gg/b4+UR1ARfoFokisCSMpVOroLJkZR6R5+apeXgqC8ZII4YiaApCLObmyi0qM7dPcJDwVBecgCMaRAAElFnuXYROXhkfIIKvIvDc1zRQBJuUpHP8HEJCq1ZcxHdyCofvZRaoXANAJIahqZyObHIioNnBjrFkkIKrIvBc0JggCSCiJN3QQZg6gkijFukWSCGkuQ3ewB1AKB8AggqfBy1iriGEQ19C2STFDarl5TIACB4QggqeFYu9lS6KIqu0XS4cPP5T0s9bKKf/v378suXrzYmD2CaoyOFSHQCQEk1QnG8CoJWVTLy8ulQ9HPnv0gF9T58x/nCTl9+t1s5847878rV67UThKCqo2MFSDQOQEk1TnScCoMWVQaiq5HyxeL5KRelElKnx079nI+T9M6BUHVocWyEOiPAJLqj20QNYcqqgMHHl8zFL1MUjZPhwOrFgRVlRTLQaB/Akiqf8butxCiqDQUXaIqFhNSsSd18uSrtXpSCKpIlNcQGJ8Akho/By4iCE1UkokO7WlqxSRlh/b0XuektmxZyKqek9IQc0bxGVGmEBifAJIaPwduIghNVJND0U1Sk6P7ij2rWbBDa/+stvAZBGIhgKRiyWRH7Qjph/ro0SOr7opukqoqpSKykNpdjJvXEIidAJKKPcMN2hfKD/bkXdGbSiqU9jZIJatAIHgCSCr4FPbTgFB+uPXoDslKRddF6VCfrpeqWkJpZ9X2sBwEYiOApGLLaIftCeEHXDHqDhSTd5yoMuQ8hPZ1mE6qgkCQBJBUkGkbLmjvP+RnzryXj8arS8R7u+q2h+UhECsBJBVrZjtsl/cf9LK7T8xqvvf2zIqdzyCQGgEklVrGG7bX8w972d0npjXTczumxcx8CKRMAEmlnP2abff6A1/1QYhe46+ZBhaHQFIEkFRS6W7fWI8/9GV3n5hsqce4J2PkPQQgsJYAklrLhDlzCHj8wdchP/WoyorHeMviZB4EILCWAJJay4Q5FQh4++Evu+GsmuEtzgpoWQQCECgQQFIFGLysR8CTAOyQ39LSH67fdNZTfPXIsjQEIGAEkJSRYNqIgBcRqCdVvLHsgw9+i7uZN8ooK0HAFwEk5SsfQUYztqj0hN6ioOz13/9+KUieBA0BCKwQQFIrLHjVgsCYolpeXi6VlOZTIACBsAkgqbDz5yr6sUQ1rSdVfCCiK1AEAwEIVCaApCqjYsEqBMYQle6CvmHD+lW9qWnD0au0gWUgAAE/BJCUn1xEE8mQopKgNm68Kdu+fVv237//fXbixPFcVh999GE0PGkIBFImgKRSzn6PbR9CVEVB/e/ycvanP/0p/7v99q9mTzzxeI+to2oIQGAoAkhqKNIJbqdPUUlQN9+8IZOQioKSqH78zDPZpk0bEyROkyEQHwEkFV9OXbWoD1FpQMSWLZuz227bvEZQ1ptav/4L2Tvv/M4VC4KBAATqE0BS9ZmxRk0CXYqqiqAkqt27d2Vf+9rOmpGyOAQg4I0AkvKWkUjj6UJUswT1q1+dyJ750Y+un5eyARSR4qRZEEiGAJJKJtXjN7SNqExQOg91+t13r8tIvSZdtKtDfz/96eFV83Ve6qmnfjB+w4kAAhBoTABJNUbHik0INBHVLEFJUk8cOJAPO//FL15aJal//973si1bFpqEyToQgIATAkjKSSJSCqOuqDSC74tfXJe9eerUKglJUOo9maR+8/rrqz5///33uWYqpR2LtkZJAElFmVb/jaoqKg1+mCaod995JxeUzkfpprJ6L3EV/+6+e2d23327/AMhQghAoJQAkirFwswhCMwT1SxBSUT3339ffj5KvSlJ6n+WllYJSsvYAAruiD5ERtkGBLongKS6Z0qNNQhME9U8QWkknz2Sw6bFHlTx9ebNtzKAokZOWBQCngggKU/ZSDSWSVHdc8/Xpx7ik3wmh5vrnNTOnXet6UWZqDSAQjegpUAAAuERQFLh5SzKiE1U3/zmP2e6W0TZIAlJR4f09j7yyCoh6b0O/ZmUyqY6r6VDfxQIQCAsAkgqrHxFHe29996TH8I7cuTFUuFoiLl6TLouykRk10jNk9TDDz2U3XHHV6PmR+MgECMBJBVjVgNsk57/tLBwS7Zt2z/mN4fV8HETkabqLdm5J+tJSVo2z6bFdYqvGY4e4E5ByBDIsgxJsRuMTsAEpTubq2jQhO4WMSmqonSavNZwdNVNgQAEwiGApMLJVZSRTgrKGtmHqN56882858VwdKPMFAL+CSAp/zmKNsJpgrIG9yEq3b3ikUf+xTbBFAIQcE4ASTlPUKzhzROUtbtrUb34s5/lvSmrnykEIOCbAJLynZ8oo6sqKGt816Li4l4jyxQC/gkgKf85iirCuoKyxncpKj1enot7jSxTCPgmgKR85yeq6JoKyiB0Kaqbbvoit0oysEwh4JgAknKcnJhCaysoY9GVqLhVkhFlCgHfBJCU7/xEEV1XgjIYXYlKt0riyb1GlSkEfBJAUj7zEk1UXQvKwHQhKnpTRpMpBPwSQFJ+cxN8ZH0JysB0ISp6U0aTKQR8EkBSPvMSfFR9C8oAtRUVvSkjyRQCPgkgKZ95CTqqoQRlkNqKit6UkWQKAX8EkJS/nAQd0dCCMlhtREVvyigyhYA/AkjKX06CjWgsQRmwNqKiN2UUmULAFwEk5SsfwUYztqAMXFNR0Zsygkwh4IsAkvKVjyCj8SIog9dUVHqGFddNGUWmEPBBAEn5yEOwUXgTlIFsIiq7QzrPmzKKTCEwPgEkNX4Ogo3Aq6AMaBNR6Q7pe/bwvCljyBQCYxNAUmNnINDtexeUYa0rKnpTRo4pBHwQQFI+8hBUFKEIyqDWFdXdd+/M7rtvl63OFAIQGJEAkhoRfoiblqDWrft8try8HFT4dUT11ptv5m386KMPg2ojwUIgRgJIKsas9tQmE5SmIZY6otq9e1d2xx1fDbGZxAyBqAggqajS2V9jQheUkakqqvfffz9bv/4L2YkTx21VphCAwAgEkNQI0EPbZCyCMu5VRfXYo49munaKAgEIjEcASY3HPogtxyYog15VVDxm3ogxhcA4BJDUONyD2GqsgjL4VUTFkHSjxRQC4xBAUuNwd7/V2AVlCagiqs2bFxiSbsCYQmBgAkhqYOAhbO7ChQv5EGyJKoUyT1Q2JP2dd36XAg7aCAFXBJCUq3SMH4wEtbBwS5aKoIz4PFE9/NBD2ZYtC7Y4UwhAYCACSGog0CFsJlVBWW5miUpD0nnmlJFiCoHhCCCp4Vi73lLqgrLkzBLVj595Jj8Myl3SjRZTCPRPAEn1z9j9FhDU6hTNEhWDKFaz4h0E+iaApPom7Lx+BFWeoGmiYhBFOS/mQqAvAkiqL7IB1IugZidpmqg0iII7Ucxmx6cQ6IoAkuqKZGD1IKhqCZsmKkmKhyNWY8hSEGhDAEm1oRfougiqXuLKRGV3ouBxHvVYsjQE6hJAUnWJBb48gmqWwDJR6XEeXDvVjCdrQaAqASRVlVQEyyGodkmcFJVdO/XEE4+3q5i1IQCBqQSQ1FQ0cX2AoLrJ56SoOOzXDVdqgcA0AkhqGpmI5iOobpM5KSod9tP1UxQIQKB7Akiqe6auakRQ/aSjKCoO+/XDmFohIAJIKuL9AEH1m9yiqDjs1y9rak+XAJKKNPcIapjEFkXFaL9hmLOVtAggqQjzjaCGTaqJ6u233srvRNHlRb7Hjr2cD3Nft+7z+c1t9+/fl50+/e6wDWRrEBiRAJIaEX4fm0ZQfVCdX6eJ6siRF3OZtH1A4pUrV7IHHrg//zt//uPrATz99A/z+iUvCgRSIICkIsoygho3mSaqR/bsyTZsWJ+1eaTH4uJ38h6UZDVZ9Jl6VkV5TS7DewjEQgBJRZJJBOUjkSaqL31pU6bXTYrkIwmp11RWzp79IP9csqJAIHYCSCqCDCMoX0mUnG6+eUMukqee+kHt4HQoT5KadUhPt2Pilky10bJCgASQVIBJK4aMoIo0/LyWqGywQ93zU4cPP1dJUqq/7HCgHwpEAoH2BJBUe4aj1YCgRkNfacMmqo0bb6p1foqeVCW8LJQIASQVaKIRVBiJM1HdddedlQOed87p4sWLeU9Lw9EpEIidAJIKMMMmqBMnjgcYfXoh79x5Zy6Vxx57tHLjNfxc55wkpMlihwMlMwoEYieApALLsAnqySe/H1jkaYe7Y8e2XFQ/+cl/VAIhOUluklXx4l0TlKYUCKRAAEkFlGUEFVCySkK9/fZ/yEX1m9+8XvLp2lkaFGHXRNkgDInr5MlX1y7MHAhESgBJBZJYBBVIouaEedttm3NR/eUvf5mz5MrHum7KDhnSg1rhwqs0CCCpAPKMoAJIUo0Qb731S9n69V/I6oiq2Kvi/n01YLNo8ASQlPMUIijnCWoYnonqpZd+nh048Hh26NCz2aVLf5tbm85VqTfFxbxzUbFAJASQlONEIijHyWkZmu7rp96UnWvSdGHhluzq1asta2Z1CMRFAEk5zad+rPSjxSg+pwnqIKyioOz1G2/8toOaqQIC8RBAUg5zKUHt3v0NBOUwN12GZGIqTrn2rUvC1BUDASTlLIsIyllCegxHveSioHT4749//KjHLVI1BMIjgKQc5QxBOUrGAKEo3xowsWfPw/ngie3bt+ZP9m3zHKoBwmYTEBiUAJIaFPf0jSGo6WxS+uTuu+9CVCklnLbOJYCk5iLqfwEE1T/jkLaAqELKFrH2TQBJ9U14Tv0Iag6gRD9GVIkmnmavIYCk1iAZbgaCGo51iFuSqDZsWJ99+OH/hRg+MUOgEwJIqhOM9StBUPWZpbjG17/+T4gqxcTT5usEkNR1FMO9QFDDsY5hSw8++K1cVG+//XYMzaENEKhFAEnVwtV+YQTVnmGKNZiofv3r/0qx+bQ5YQJIasDkI6gBYUe4qW9/+7G8R/Xd7/5bhK2jSRAoJ4Ckyrl0PhdBdY40yQolqE2bNmb33ntPpbumJwmJRkdFAEkNkM6ioBYXF7Mbbvhs/nfu3Ll865cvX87ff/LJXzuLxrbx/PMv5HVqqnl79+7rbBtUNA4B3U7pK1+5Lbv11luyM2feGycItgqBgQggqZ5BFwVlm5KcJAwTiCS1deu2bGlpyRbpZHrw4MG8Xm3nlVeOd1InlfggIFHdeeeOXFS6tZL2MwoEYiSApHrMapmgbHNFSWmeeliS1WTZtGlTLjTrGU1Otd60Yj20WctMW5f5/glIVBqirkEVO3Zsz5aXl/0HTYQQqEkASdUEVnXxWYJSHZKPyUOH+axXVbX+qsvdeOPnslOnTlVdnOUCIyBR6bEuv/zlf+bPH6NXFVgCCXcuASQ1F1H9BeYJSjVKUnZ+SNOyXlT9La9dQ9vRYT9KvARMVH/+85/zO6qrV8W5qnjznVrLkFTHGa8iKG1y167d+Z96ObPOF7U53KfemXprOt9FiZuAiUr7nwSlpzrrESCXLv0t7obTuugJIKkOU1xVUNqkek/FQ34dhpFXpcEZ2oYN0tAhRQmrrx5b1/FTX30CRVFpX9ShPz1U8ejRIwysqI+TNZwQQFIdJaKOoLRJCUPni7ocdm5N0eAK9Z5MSHqtbXU9etC2x9QPgaKoFJV6UgcOPJ73rJCVnzwRSXUCSKo6q6lL1hWUKpKkGNAwFSkftCAwKSpVpZF/Ovyn81VvvPHbFrWzKgSGJYCkWvJuIijJicEMLcGz+kwCZaLSCpOy0v5LgYBnAkiqRXbqCkrnoGwwQ4vNsioEKhGYJiqtbLLSAAsOA1bCyUIjEUBSDcHXFZTOD+m8kEb1USAwFIFZolIMkpWds9KyjAYcKjNspyoBJFWVVGE5E5QuoqRAwDuBeaJS/JKTlrOh61xn5T2r6cSHpGrmuigovaZAIAQCVUSldmif1sAKDbDQn4ax07sKIcPxxoikauQWQdWAxaLuCFQVlQWuQ4HF3pXkxT9mRofpUASQVEXSCKoiKBZzTaCuqNQY611pCLsuDlYdHA50neaogkNSFdKJoCpAYpFgCDQRlTVOh/5OnDie39RW568QlpFh2hcBJDWHLIKaA4iPgyTQRlTW4KKw1MPSKEEOCRodpl0RQFIzSCKoGXD4KHgCXYjKIJiwJCoJSyNfdf3VhQsXbBGmEGhEAEnNwNbll3jGZvgIAqMR6GMf1z93OmelujVC0A4LqpfFSMHRUh3shpHUlNT18eWdsilmQ2BUAn3v6+pN6TyW9bIkLm0TaY2a9mA2jqRKUtX3l7Zkk8yCwKgEhtznNbRdhwJttKBJSyLj8OCou4HLjSOpibQM+WWd2DRvITAqgbH2/WJPS8LSOS0JTCLTYUMOEY66W4y+cSRVSMFYX9JCCLyEwKgEPHwHJCXJqdjbsts1Ia5Rd49RNo6kPsXu4cs5yh7ARiEwQcDjd0G9LZ3DMnFJWtbjUrw6VKjDiBq0QYmLAJLKsvwkrobMsoPHtXPTmuYEPIpqsjX6vtr5LcVr57iK8rKeF+e6Juldu5OI+HgvyUsqhC+j952I+OIkEOp3Q4cLTV66Qa7kZee6NNV7tc0ElmoPTAwkdE09/4OetKRC/RLG+ZNIqzwSiO07oh6VCcwOHeooin6s9afXRYnZYcQYRSYxmag8H0lKVlKxffk8/sARUxwEUvquSEZFienaLkmrKDIbxKH56qlJdsVeWWhCqyIqPaz1hhs+m+3duy/fqZeWlvKHuOpBrn2XJCWV0peu7x2I+tMgwHdmJc/WG7MRiBKUHVacFJp6Z0WpFXtpJjcNCDE52nTow29VRHXq1KlcVBLU4uLiCpCeXyUpKc9d257zTfUQaEzARNW4gkRXNKlJQEWxmaQkruJfsddmhyFtqs+Ky06+tjqbTFWXtrNx403Za6+9VpqtrVu3ZZs2bSr9rK+Zg0nKIDO9duwbDnBgH2Af8LoPvPTSz0udo8N9Bw8eLP2sr5mDSaqvBtStVzsFBQIQgAAEVgjokKN+G3VoctZwfR3mU29qyIKkhqTNtiAAAQg4I1BVUOfOncvPRWkAxeXLlwdrBZIaDDUbggAEIOCLgIbYV+lBSUrqQWmqEX2vvHI8/9Mgir4LkuqbMPVDAAIQcEhAIxIlKA3GmHUTXw0/l5jUk1LROSn1pp5//oVBWoWkBsHMRiAAAQj4IqBzT7oObOjh7nUpIKm6xFgeAhCAAAQGI4CkBkPNhiAAAQhAoC4BJFWXGMtDAAIQgMBgBJKT1GBk2RAEIAABCLQmgKRaI6QCCEAAAhDoiwCS6oss9UIAAhCAQGsCSKo1QiqAAAQgAIG+CCCpvshSLwQgAAEItCaApFojpAIIQAACEOiLAJLqiyz1QgACEPLtJSsAAAA/SURBVIBAawJIqjVCKoAABCAAgb4IIKm+yFIvBCAAAQi0JoCkWiOkAghAAAIQ6IsAkuqLLPVCAAIQgEBrAv8PXrSFimBzMx8AAAAASUVORK5CYII="></p>
</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 <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the exact coordinates of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Q</mtext></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>Show that the equation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mi>x</mi><mo>+</mo><mn>2</mn><mo> </mo><mi>ln</mi><mo> </mo><mn>45</mn><mo>+</mo><mn>2</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-coordinate of the point where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> intersects the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, find the area 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">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> is tangent to the graphs of both <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> and the inverse function <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>.</p>
<p style="text-align:center;"><img src="data:image/png;base64,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"></p>
<p>Find the shaded area enclosed by the graphs of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> and the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = \ln x">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>ln</mi>
<mo><!-- --></mo>
<mi>x</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g(x) = 3 + \ln \left( {\frac{x}{2}} \right)">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>3</mn>
<mo>+</mo>
<mi>ln</mi>
<mo><!-- --></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>x</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x > 0">
<mi>x</mi>
<mo>></mo>
<mn>0</mn>
</math></span>.</p>
<p>The graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g">
<mi>g</mi>
</math></span> can be obtained from the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> by two transformations:</p>
<p><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="\begin{array}{*{20}{l}} {{\text{a horizontal stretch of scale factor }}q{\text{ followed by}}} \\ {{\text{a translation of }}\left( {\begin{array}{*{20}{c}} h \\ k \end{array}} \right).} \end{array}">
<mtable columnalign="left" rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mrow>
<mtext>a horizontal stretch of scale factor </mtext>
</mrow>
<mi>q</mi>
<mrow>
<mtext> followed by</mtext>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mrow>
<mtext>a translation of </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>h</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>k</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>.</mo>
</mrow>
</mtd>
</mtr>
</mtable>
</math></span></p>
</div>
<div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h(x) = g(x) \times \cos (0.1x)">
<mi>h</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>×<!-- × --></mo>
<mi>cos</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mn>0.1</mn>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span>, for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 < x < 4">
<mn>0</mn>
<mo><</mo>
<mi>x</mi>
<mo><</mo>
<mn>4</mn>
</math></span>. The following diagram shows the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h">
<mi>h</mi>
</math></span> and the line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = x">
<mi>y</mi>
<mo>=</mo>
<mi>x</mi>
</math></span>.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-14_om_10.34.27.png" alt="M17/5/MATME/SP2/ENG/TZ1/10.b.c"></p>
<p>The graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h">
<mi>h</mi>
</math></span> intersects the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{h^{ - 1}}">
<mrow>
<msup>
<mi>h</mi>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span> at two points. These points have <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> coordinates 0.111 and 3.31 correct to three significant figures.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q"> <mi>q</mi> </math></span>;</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>Write down the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h"> <mi>h</mi> </math></span>;</p>
<div class="marks">[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 value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k"> <mi>k</mi> </math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.iii.</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="\int_{0.111}^{3.31} {\left( {h(x) - x} \right){\text{d}}x} "> <msubsup> <mo>∫</mo> <mrow> <mn>0.111</mn> </mrow> <mrow> <mn>3.31</mn> </mrow> </msubsup> <mrow> <mrow> <mo>(</mo> <mrow> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, find the area of the region enclosed by the graphs of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h"> <mi>h</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{h^{ - 1}}"> <mrow> <msup> <mi>h</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d"> <mi>d</mi> </math></span> be the vertical distance from a point on the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h"> <mi>h</mi> </math></span> to the line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = x"> <mi>y</mi> <mo>=</mo> <mi>x</mi> </math></span>. There is a point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(a,{\text{ }}b)"> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>b</mi> <mo stretchy="false">)</mo> </math></span> on the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h"> <mi>h</mi> </math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d"> <mi>d</mi> </math></span> is a maximum.</p>
<p>Find the coordinates of P, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0.111 < a < 3.31"> <mn>0.111</mn> <mo><</mo> <mi>a</mi> <mo><</mo> <mn>3.31</mn> </math></span>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Gemma and Kaia started working for different companies on January 1st 2011.</p>
<p>Gemma’s starting annual salary was <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>$</mo><mn>45</mn><mo> </mo><mn>000</mn></math>, and her annual salary increases <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>%</mo></math> on January 1st each year after 2011.</p>
</div>
<div class="specification">
<p>Kaia’s annual salary is based on a yearly performance review. Her salary for the years 2011, 2013, 2014, 2018, and 2022 is shown in the following table.</p>
<p style="text-align: left;"><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find Gemma’s annual salary for the year 2021, to the nearest dollar.</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>Assuming Kaia’s annual salary can be approximately modelled by the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mo>=</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi></math>, show that Kaia had a higher salary than Gemma in the year 2021, according to the model.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A rocket is travelling in a straight line, with an initial velocity of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="140">
<mn>140</mn>
</math></span> m s<sup>−1</sup>. It accelerates to a new velocity of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="500">
<mn>500</mn>
</math></span> m s<sup>−1</sup> in two stages.</p>
<p>During the first stage its acceleration, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> m s<sup>−2</sup>, after <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> seconds is given by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a\left( t \right) = 240\,{\text{sin}}\left( {2t} \right)">
<mi>a</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>240</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>t</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 \leqslant t \leqslant k">
<mn>0</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>t</mi>
<mo>⩽<!-- ⩽ --></mo>
<mi>k</mi>
</math></span>.</p>
</div>
<div class="specification">
<p>The first stage continues for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span> seconds until the velocity of the rocket reaches <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="375">
<mn>375</mn>
</math></span> m s<sup>−1</sup>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for the velocity, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v"> <mi>v</mi> </math></span> m s<sup>−1</sup>, of the rocket during the first stage.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the distance that the rocket travels during the first stage.</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>During the second stage, the rocket accelerates at a constant rate. The distance which the rocket travels during the second stage is the same as the distance it travels during the first stage.</p>
<p>Find the total time taken for the two stages.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = {x^2} - 1">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g(x) = {x^2} - 2">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>2</mn>
</math></span>, for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \in \mathbb{R}">
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>.</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="(f \circ g)(x) = {x^4} - 4{x^2} + 3"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∘</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <msup> <mi>x</mi> <mn>4</mn> </msup> </mrow> <mo>−</mo> <mn>4</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>3</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>On the following grid, sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(f \circ g)(x)"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∘</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span>, for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 \leqslant x \leqslant 2.25"> <mn>0</mn> <mo>⩽</mo> <mi>x</mi> <mo>⩽</mo> <mn>2.25</mn> </math></span>.</p>
<p><img src="images/Schermafbeelding_2017-08-15_om_08.00.33.png" alt="M17/5/MATME/SP2/ENG/TZ2/06.b"></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>The equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(f \circ g)(x) = k"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∘</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>k</mi> </math></span> has exactly two solutions, for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 \leqslant x \leqslant 2.25"> <mn>0</mn> <mo>⩽</mo> <mi>x</mi> <mo>⩽</mo> <mn>2.25</mn> </math></span>. Find the possible values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k"> <mi>k</mi> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = \frac{{6x - 1}}{{2x + 3}}">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>6</mn>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>3</mn>
</mrow>
</mfrac>
</math></span>, for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \ne - \frac{3}{2}">
<mi>x</mi>
<mo>≠<!-- ≠ --></mo>
<mo>−<!-- − --></mo>
<mfrac>
<mn>3</mn>
<mn>2</mn>
</mfrac>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>For the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span>, find the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-intercept.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence or otherwise, write down <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{lim}}}\limits_{x \to \infty } \left( {\frac{{6x - 1}}{{2x + 3}}} \right)">
<munder>
<mrow>
<mrow>
<mtext>lim</mtext>
</mrow>
</mrow>
<mrow>
<mi>x</mi>
<mo stretchy="false">→</mo>
<mi mathvariant="normal">∞</mi>
</mrow>
</munder>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>6</mn>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>3</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The following diagram shows the graph of a function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f(x)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span>, for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 6 \leqslant x \leqslant - 2">
<mo>−<!-- − --></mo>
<mn>6</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</mi>
<mo>⩽<!-- ⩽ --></mo>
<mo>−<!-- − --></mo>
<mn>2</mn>
</math></span>.</p>
<p>The points <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="( - 6,{\text{ }}6)">
<mo stretchy="false">(</mo>
<mo>−<!-- − --></mo>
<mn>6</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>6</mn>
<mo stretchy="false">)</mo>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="( - 2,{\text{ }}6)">
<mo stretchy="false">(</mo>
<mo>−<!-- − --></mo>
<mn>2</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>6</mn>
<mo stretchy="false">)</mo>
</math></span> lie on the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span>. There is a minimum point at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="( - 4,{\text{ }}0)">
<mo stretchy="false">(</mo>
<mo>−<!-- − --></mo>
<mn>4</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>0</mn>
<mo stretchy="false">)</mo>
</math></span>.</p>
<p><img src="data:image/png;base64,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"></p>
</div>
<div class="specification">
<p style="text-align: left;">Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g(x) = f(x - 5)">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>5</mn>
<mo stretchy="false">)</mo>
</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">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the domain 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="specification">
<p>A discrete random variable, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math>, has the following probability distribution:</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msup><mi>k</mi><mn>2</mn></msup><mo>-</mo><mi>k</mi><mo>+</mo><mn>0</mn><mo>.</mo><mn>12</mn><mo>=</mo><mn>0</mn></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>k</mi></math>, giving a reason for your answer.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, find <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>E</mtext><mo>(</mo><mi>X</mi><mo>)</mo></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = \frac{{8x - 5}}{{cx + 6}}">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>8</mn>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>5</mn>
</mrow>
<mrow>
<mi>c</mi>
<mi>x</mi>
<mo>+</mo>
<mn>6</mn>
</mrow>
</mfrac>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \ne - \frac{6}{c},\,\,c \ne 0">
<mi>x</mi>
<mo>≠<!-- ≠ --></mo>
<mo>−<!-- − --></mo>
<mfrac>
<mn>6</mn>
<mi>c</mi>
</mfrac>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>c</mi>
<mo>≠<!-- ≠ --></mo>
<mn>0</mn>
</math></span>.</p>
</div>
<div class="question">
<p>Write down the equation of the horizontal asymptote to the graph of <em>f</em>.</p>
</div>
<br><hr><br><div class="specification">
<p>Let <em>f</em>(<em>x</em>) = ln <em>x</em> − 5<em>x</em> , for <em>x</em> > 0 .</p>
</div>
<div class="question">
<p>Solve<em> f '</em>(<em>x</em>)<em> = f "</em>(<em>x</em>).</p>
</div>
<br><hr><br>