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<h2>HL Paper 2</h2><div class="specification">
<p>A function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mtext>arcsin</mtext><mfenced><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced><mo>,</mo><mo> </mo><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
</div>
<div class="specification">
<p>A function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mtext>arcsin</mtext><mfenced><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced><mo>,</mo><mo> </mo><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>≥</mo><mn>0</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is an even function.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By considering limits, show that the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> has a horizontal asymptote and state its equation.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><msqrt><msup><mi>x</mi><mn>2</mn></msup></msqrt><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfrac></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>≠</mo><mn>0</mn></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By using the expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced></math> and the result <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><msup><mi>x</mi><mn>2</mn></msup></msqrt><mo>=</mo><mfenced open="|" close="|"><mi>x</mi></mfenced></math>, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is decreasing for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo><</mo><mn>0</mn></math>.</p>
<p> </p>
<div class="marks">[3]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo></math>, justifying your answer.</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the domain of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo></math>, clearly indicating any asymptotes with their equations and stating the values of any axes intercepts.</p>
<div class="marks">[3]</div>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cot}}\,2\theta = \frac{{1 - {\text{ta}}{{\text{n}}^2}\,\theta }}{{2\,{\text{tan}}\,\theta }}">
<mrow>
<mtext>cot</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>θ</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<mtext>ta</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</mrow>
<mrow>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</mrow>
</mfrac>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Verify that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = {\text{tan}}\,\theta ">
<mi>x</mi>
<mo>=</mo>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = - \,{\text{cot}}\,\theta ">
<mi>x</mi>
<mo>=</mo>
<mo>−</mo>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cot</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</math></span> satisfy the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} + \left( {2\,{\text{cot}}\,2\theta } \right)x - 1 = 0">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cot</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>θ</mi>
</mrow>
<mo>)</mo>
</mrow>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
<mo>=</mo>
<mn>0</mn>
</math></span>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, or otherwise, show that the exact value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\frac{\pi }{{12}} = 2 - \sqrt 3 ">
<mrow>
<mtext>tan</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mn>2</mn>
<mo>−</mo>
<msqrt>
<mn>3</mn>
</msqrt>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using the results from parts (b) and (c) find the exact value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\frac{\pi }{{24}} - {\text{cot}}\frac{\pi }{{24}}">
<mrow>
<mtext>tan</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>24</mn>
</mrow>
</mfrac>
<mo>−</mo>
<mrow>
<mtext>cot</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>24</mn>
</mrow>
</mfrac>
</math></span>.</p>
<p>Give your answer in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a + b\sqrt 3 ">
<mi>a</mi>
<mo>+</mo>
<mi>b</mi>
<msqrt>
<mn>3</mn>
</msqrt>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b \in \mathbb{Z}">
<mi>b</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
</math></span>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>A continuous random variable <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> has a probability density function given by</p>
<p style="padding-left: 180px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced open="{" close><mtable><mtr><mtd><mtext>arccos</mtext><mo> </mo><mi>x</mi><mo> </mo></mtd><mtd><mn>0</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mtext>otherwise</mtext></mtd></mtr></mtable></mfenced></math></p>
<p>The median of this distribution is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mfenced><mrow><mfenced open="|" close="|"><mrow><mi>X</mi><mo>-</mo><mi>m</mi></mrow></mfenced><mo>≤</mo><mi>a</mi></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>3</mn></math>, determine the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The height of water, in metres, in Dungeness harbour is modelled by the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mi>a</mi><mo> </mo><mi>sin</mi><mo>(</mo><mi>b</mi><mo>(</mo><mi>t</mi><mo>-</mo><mi>c</mi><mo>)</mo><mo>)</mo><mo>+</mo><mi>d</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> is the number of hours after midnight, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>,</mo><mo> </mo><mi>c</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math> are constants, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>></mo><mn>0</mn><mo>,</mo><mo> </mo><mi>b</mi><mo>></mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>></mo><mn>0</mn></math>.</p>
<p>The following graph shows the height of the water for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>13</mn></math> hours, starting at midnight.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p>The first high tide occurs at <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>04</mn><mo>:</mo><mn>30</mn></math> and the next high tide occurs <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn></math> hours later. Throughout the day, the height of the water fluctuates between <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>2</mn><mo> </mo><mtext>m</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>.</mo><mn>8</mn><mo> </mo><mtext>m</mtext></math>.</p>
<p>All heights are given correct to one decimal place.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the smallest possible value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the height of the water at <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn><mo>:</mo><mn>00</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the number of hours, over a 24-hour period, for which the tide is higher than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math> metres.</p>
<div class="marks">[3]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>A fisherman notes that the water height at nearby Folkestone harbour follows the same sinusoidal pattern as that of Dungeness harbour, with the exception that high tides (and low tides) occur <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>50</mn></math> minutes earlier than at Dungeness.</p>
<p>Find a suitable equation that may be used to model the tidal height of water at Folkestone harbour.</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.</div>
</div>
<br><hr><br><div class="specification">
<p>The voltage <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v">
<mi>v</mi>
</math></span> in a circuit is given by the equation</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v\left( t \right) = 3\,{\text{sin}}\left( {100\pi t} \right)">
<mi>v</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>100</mn>
<mi>π<!-- π --></mi>
<mi>t</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t \geqslant 0">
<mi>t</mi>
<mo>⩾<!-- ⩾ --></mo>
<mn>0</mn>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> is measured in seconds.</p>
</div>
<div class="specification">
<p>The current <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="i">
<mi>i</mi>
</math></span> in this circuit is given by the equation</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="i\left( t \right) = 2\,{\text{sin}}\left( {100\pi \left( {t + 0.003} \right)} \right)">
<mi>i</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>100</mn>
<mi>π<!-- π --></mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>+</mo>
<mn>0.003</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>The power <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span> in this circuit is given by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\left( t \right) = v\left( t \right) \times i\left( t \right)">
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>v</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>×<!-- × --></mo>
<mi>i</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>The average power <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{p_{av}}">
<mrow>
<msub>
<mi>p</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
</mrow>
</msub>
</mrow>
</math></span> in this circuit from <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 0">
<mi>t</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = T">
<mi>t</mi>
<mo>=</mo>
<mi>T</mi>
</math></span> is given by the equation</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{p_{av}}\left( T \right) = \frac{1}{T}\int_0^T {p\left( t \right){\text{d}}t} ">
<mrow>
<msub>
<mi>p</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mi>T</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mi>T</mi>
</mfrac>
<msubsup>
<mo>∫<!-- ∫ --></mo>
<mn>0</mn>
<mi>T</mi>
</msubsup>
<mrow>
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="T > 0">
<mi>T</mi>
<mo>></mo>
<mn>0</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the maximum and minimum value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v">
<mi>v</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down two transformations that will transform the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = v\left( t \right)">
<mi>y</mi>
<mo>=</mo>
<mi>v</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> onto the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = i\left( t \right)">
<mi>y</mi>
<mo>=</mo>
<mi>i</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = p\left( t \right)">
<mi>y</mi>
<mo>=</mo>
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> for 0 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> ≤ 0.02 , showing clearly the coordinates of the first maximum and the first minimum.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the total time in the interval 0 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> ≤ 0.02 for which <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\left( t \right)">
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> ≥ 3.</p>
<p> </p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{p_{av}}">
<mrow>
<msub>
<mi>p</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
</mrow>
</msub>
</mrow>
</math></span>(0.007).</p>
<p> </p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>With reference to your graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = p\left( t \right)">
<mi>y</mi>
<mo>=</mo>
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> explain why <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{p_{av}}\left( T \right)">
<mrow>
<msub>
<mi>p</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mi>T</mi>
<mo>)</mo>
</mrow>
</math></span> > 0 for all <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="T">
<mi>T</mi>
</math></span> > 0.</p>
<p> </p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\left( t \right)">
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> can be written as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\left( t \right) = a\,{\text{sin}}\left( {b\left( {t - c} \right)} \right) + d">
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>a</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>b</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>−</mo>
<mi>c</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>d</mi>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c">
<mi>c</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d">
<mi>d</mi>
</math></span> > 0, use your graph to find the values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c">
<mi>c</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d">
<mi>d</mi>
</math></span>.</p>
<p> </p>
<div class="marks">[6]</div>
<div class="question_part_label">g.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the rectangle OABC such that AB = OC = 10 and BC = OA = 1 , with the points P , Q and R placed on the line OC such that OP = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span>, OQ = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q">
<mi>q</mi>
</math></span> and OR = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span>, such that 0 < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span> < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q">
<mi>q</mi>
</math></span> < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span> < 10.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\theta _p}">
<mrow>
<msub>
<mi>θ<!-- θ --></mi>
<mi>p</mi>
</msub>
</mrow>
</math></span> be the angle APO, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\theta _q}">
<mrow>
<msub>
<mi>θ<!-- θ --></mi>
<mi>q</mi>
</msub>
</mrow>
</math></span> be the angle AQO and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\theta _r}">
<mrow>
<msub>
<mi>θ<!-- θ --></mi>
<mi>r</mi>
</msub>
</mrow>
</math></span> be the angle ARO.</p>
</div>
<div class="specification">
<p>Consider the case when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\theta _p} = {\theta _q} + {\theta _r}">
<mrow>
<msub>
<mi>θ<!-- θ --></mi>
<mi>p</mi>
</msub>
</mrow>
<mo>=</mo>
<mrow>
<msub>
<mi>θ<!-- θ --></mi>
<mi>q</mi>
</msub>
</mrow>
<mo>+</mo>
<mrow>
<msub>
<mi>θ<!-- θ --></mi>
<mi>r</mi>
</msub>
</mrow>
</math></span> and QR = 1.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\theta _p}"> <mrow> <msub> <mi>θ</mi> <mi>p</mi> </msub> </mrow> </math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p = \frac{{{q^2} + q - 1}}{{2q + 1}}"> <mi>p</mi> <mo>=</mo> <mfrac> <mrow> <mrow> <msup> <mi>q</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mi>q</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <mi>q</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </math></span>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By sketching the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span> as a function of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q"> <mi>q</mi> </math></span>, determine the range of values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span> for which there are possible values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q"> <mi>q</mi> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The plane <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Π</mi><mn>1</mn></msub></math> has equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>x</mi><mo>−</mo><mi>y</mi><mo>+</mo><mi>z</mi><mo>=</mo><mo>−</mo><mn>13</mn></math> and the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> has vector equation</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>λ</mi><mfenced><mtable><mtr><mtd><mo>-</mo><mn>3</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>4</mn></mtd></mtr></mtable></mfenced><mo> </mo><mo>,</mo><mo> </mo><mi>λ</mi><mo> </mo><mo>∈</mo><mo> </mo><mi mathvariant="normal">ℝ</mi></math>.</p>
</div>
<div class="specification">
<p>The plane <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Π</mi><mn>2</mn></msub></math> contains the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math> and the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> meets <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Π</mi><mn>1</mn></msub></math> at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>, find the coordinates of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>.</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 shortest distance from the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Π</mi><mn>1</mn></msub></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Π</mi><mn>2</mn></msub></math>, giving your answer in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo mathvariant="bold">.</mo><mi mathvariant="bold-italic">n</mi><mo>=</mo><mi>d</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the acute angle between <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Π</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Π</mi><mn>2</mn></msub></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>A particle <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> moves in a straight line such that after time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> seconds, its velocity, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi></math> in <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>m s</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>, is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>=</mo><msup><mtext>e</mtext><mrow><mo>−</mo><mn>3</mn><mi>t</mi></mrow></msup><mo> </mo><mi>sin</mi><mo> </mo><mn>6</mn><mo> </mo><mi>t</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo><</mo><mi>t</mi><mo><</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></math>.</p>
</div>
<div class="specification">
<p>At time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> has displacement <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>(</mo><mi>t</mi><mo>)</mo></math>; at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mn>0</mn></math>.</p>
</div>
<div class="specification">
<p>At successive times when the acceleration of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> is<math xmlns="http://www.w3.org/1998/Math/MathML"><mo> </mo><mn>0</mn><mo> </mo><msup><mtext>m s</mtext><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo> </mo></math>, the velocities of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> form a geometric sequence. The acceleration of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> is zero at times <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>1</mn></msub><mo>,</mo><mo> </mo><msub><mi>t</mi><mn>2</mn></msub><mo>,</mo><mo> </mo><msub><mi>t</mi><mn>3</mn></msub></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>1</mn></msub><mo><</mo><msub><mi>t</mi><mn>2</mn></msub><mo><</mo><msub><mi>t</mi><mn>3</mn></msub></math> and the respective velocities are <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>v</mi><mn>1</mn></msub><mo>,</mo><mo> </mo><msub><mi>v</mi><mn>2</mn></msub><mo>,</mo><mo> </mo><msub><mi>v</mi><mn>3</mn></msub></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the times when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> comes to instantaneous rest.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the maximum displacement of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math>, in metres, from its initial position.</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 total distance travelled by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> in the first <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>5</mn></math> seconds of its motion.</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>Show that, at these times, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mo> </mo><mn>6</mn><mi>t</mi><mo>=</mo><mn>2</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>v</mi><mn>2</mn></msub><msub><mi>v</mi><mn>1</mn></msub></mfrac><mo>=</mo><mfrac><msub><mi>v</mi><mn>3</mn></msub><msub><mi>v</mi><mn>2</mn></msub></mfrac><mo>=</mo><mo>-</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></mrow></msup></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">e.ii.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the set of values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k"> <mi>k</mi> </math></span> that satisfy the inequality <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{k^2} - k - 12 < 0"> <mrow> <msup> <mi>k</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mi>k</mi> <mo>−</mo> <mn>12</mn> <mo><</mo> <mn>0</mn> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The triangle ABC is shown in the following diagram. Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos B < \frac{1}{4}"> <mi>cos</mi> <mo></mo> <mi>B</mi> <mo><</mo> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </math></span>, find the range of possible values for AB.</p>
<p><img src="images/Schermafbeelding_2017-08-09_om_18.13.24.png" alt="M17/5/MATHL/HP2/ENG/TZ2/04.b"></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Two airplanes, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math>, have position vectors with respect to an origin <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math> given respectively by</p>
<p style="padding-left: 180px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">r</mi><mtext mathvariant="bold-italic">A</mtext></msub><mo>=</mo><mfenced><mtable><mtr><mtd><mn>19</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>t</mi><mfenced><mtable><mtr><mtd><mo>-</mo><mn>6</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mn>4</mn></mtd></mtr></mtable></mfenced></math></p>
<p style="padding-left: 180px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">r</mi><mi mathvariant="bold-italic">B</mi></msub><mo>=</mo><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>12</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>t</mi><mfenced><mtable><mtr><mtd><mn>4</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr></mtable></mfenced></math></p>
<p>where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> represents the time in minutes and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mn>2</mn><mo>.</mo><mn>5</mn></math>.</p>
<p>Entries in each column vector give the displacement east of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math>, the displacement north of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math> and the distance above sea level, all measured in kilometres.</p>
</div>
<div class="specification">
<p>The two airplanes’ lines of flight cross at point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the three-figure bearing on which airplane <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> is travelling.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that airplane <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> travels at a greater speed than airplane <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the acute angle between the two airplanes’ lines of flight. Give your answer in degrees.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the coordinates of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the length of time between the first airplane arriving at <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> and the second airplane arriving at <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>(</mo><mi>t</mi><mo>)</mo></math> represent the distance between airplane <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and airplane <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mn>2</mn><mo>.</mo><mn>5</mn></math>.</p>
<p>Find the minimum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>(</mo><mi>t</mi><mo>)</mo></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>A water trough which is 10 metres long has a uniform cross-section in the shape of a semicircle with radius 0.5 metres. It is partly filled with water as shown in the following diagram of the cross-section. The centre of the circle is O and the angle KOL is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta ">
<mi>θ<!-- θ --></mi>
</math></span> radians.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-09_om_11.09.30.png" alt="M17/5/MATHL/HP2/ENG/TZ1/08"></p>
</div>
<div class="specification">
<p>The volume of water is increasing at a constant rate of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0.0008{\text{ }}{{\text{m}}^3}{{\text{s}}^{ - 1}}">
<mn>0.0008</mn>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>m</mtext>
</mrow>
<mn>3</mn>
</msup>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>s</mtext>
</mrow>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for the volume of water <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="V{\text{ }}({{\text{m}}^3})">
<mi>V</mi>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mrow>
<mtext>m</mtext>
</mrow>
<mn>3</mn>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</math></span> in the trough in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta ">
<mi>θ</mi>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}\theta }}{{{\text{d}}t}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
</math></span> when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta = \frac{\pi }{3}">
<mi>θ</mi>
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mn>3</mn>
</mfrac>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = 3x\arccos (x)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>3</mn>
<mi>x</mi>
<mi>arccos</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 1 \leqslant x \leqslant 1">
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</mi>
<mo>⩽<!-- ⩽ --></mo>
<mn>1</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> indicating clearly any intercepts with the axes and the coordinates of any local maximum or minimum points.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the range of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve the inequality <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| {3x\arccos (x)} \right| > 1">
<mrow>
<mo>|</mo>
<mrow>
<mn>3</mn>
<mi>x</mi>
<mi>arccos</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mrow>
<mo>|</mo>
</mrow>
<mo>></mo>
<mn>1</mn>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = 2{\sin ^2}x + 7\sin 2x + \tan x - 9,{\text{ }}0 \leqslant x < \frac{\pi }{2}">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>2</mn>
<mrow>
<msup>
<mi>sin</mi>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
<mo>+</mo>
<mn>7</mn>
<mi>sin</mi>
<mo><!-- --></mo>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mi>tan</mi>
<mo><!-- --></mo>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>9</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>0</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</mi>
<mo><</mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>2</mn>
</mfrac>
</math></span>.</p>
</div>
<div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u = \tan x">
<mi>u</mi>
<mo>=</mo>
<mi>tan</mi>
<mo><!-- --></mo>
<mi>x</mi>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine an expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f’(x)"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch a graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f’(x)"> <mi>y</mi> <mo>=</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 \leqslant x < \frac{\pi }{2}"> <mn>0</mn> <mo>⩽</mo> <mi>x</mi> <mo><</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-coordinate(s) of the point(s) of inflexion of the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f(x)"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span>, labelling these clearly on the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f’(x)"> <mi>y</mi> <mo>=</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Express <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin x"> <mi>sin</mi> <mo></mo> <mi>x</mi> </math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu "><mi>u</mi></math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Express <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin 2x"> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> </math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u"> <mi>u</mi> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = 0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </math></span> can be expressed as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u^3} - 7{u^2} + 15u - 9 = 0"> <mrow> <msup> <mi>u</mi> <mn>3</mn> </msup> </mrow> <mo>−</mo> <mn>7</mn> <mrow> <msup> <mi>u</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>15</mn> <mi>u</mi> <mo>−</mo> <mn>9</mn> <mo>=</mo> <mn>0</mn> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = 0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </math></span>, giving your answers in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\arctan k"> <mi>arctan</mi> <mo></mo> <mi>k</mi> </math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k \in \mathbb{Z}"> <mi>k</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Two submarines A and B have their routes planned so that their positions at time <em>t</em> hours, 0 ≤ <em>t</em> < 20 , would be defined by the position vectors <em><strong>r</strong><sub>A</sub></em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( \begin{gathered} \,2 \hfill \\ \,4 \hfill \\ - 1 \hfill \\ \end{gathered} \right) + t\left( \begin{gathered} - 1 \hfill \\ \,1 \hfill \\ - 0.15 \hfill \\ \end{gathered} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>t</mi>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−<!-- − --></mo>
<mn>0.15</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span> and <em><strong>r</strong><sub>B</sub></em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( \begin{gathered} \,0 \hfill \\ \,3.2 \hfill \\ - 2 \hfill \\ \end{gathered} \right) + t\left( \begin{gathered} - 0.5 \hfill \\ \,1.2 \hfill \\ \,0.1 \hfill \\ \end{gathered} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>3.2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>t</mi>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mo>−<!-- − --></mo>
<mn>0.5</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>1.2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>0.1</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span> relative to a fixed point on the surface of the ocean (all lengths are in kilometres).</p>
</div>
<div class="specification">
<p>To avoid the collision submarine B adjusts its velocity so that its position vector is now given by</p>
<p style="padding-left: 120px;"><em><strong>r</strong><sub>B</sub></em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( \begin{gathered} \,0 \hfill \\ \,3.2 \hfill \\ - 2 \hfill \\ \end{gathered} \right) + t\left( \begin{gathered} - 0.45 \hfill \\ \,1.08 \hfill \\ \,0.09 \hfill \\ \end{gathered} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>3.2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>t</mi>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mo>−<!-- − --></mo>
<mn>0.45</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>1.08</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>0.09</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the two submarines would collide at a point P and write down the coordinates of P.</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>Show that submarine B travels in the same direction as originally planned.</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 the value of <em>t</em> when submarine B passes through P.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for the distance between the two submarines in terms of <em>t</em>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <em>t</em> when the two submarines are closest together.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the distance between the two submarines at this time.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.iii.</div>
</div>
<br><hr><br><div class="specification">
<p>The points A, B and C have the following position vectors with respect to an origin O.</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\rm{OA}}} = 2">
<mover>
<mrow>
<mrow>
<mi mathvariant="normal">O</mi>
<mi mathvariant="normal">A</mi>
</mrow>
</mrow>
<mo>→<!-- → --></mo>
</mover>
<mo>=</mo>
<mn>2</mn>
</math></span><strong><em>i</em></strong> + <strong><em>j</em></strong> – 2<strong><em>k</em></strong></p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\rm{OB}}} = 2">
<mover>
<mrow>
<mrow>
<mi mathvariant="normal">O</mi>
<mi mathvariant="normal">B</mi>
</mrow>
</mrow>
<mo>→<!-- → --></mo>
</mover>
<mo>=</mo>
<mn>2</mn>
</math></span><strong><em>i</em></strong> – <strong><em>j</em></strong> + 2<strong><em>k</em></strong></p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\rm{OC}}} = ">
<mover>
<mrow>
<mrow>
<mi mathvariant="normal">O</mi>
<mi mathvariant="normal">C</mi>
</mrow>
</mrow>
<mo>→<!-- → --></mo>
</mover>
<mo>=</mo>
</math></span> <strong><em>i</em></strong> + 3<strong><em>j</em></strong> + 3<strong><em>k</em></strong></p>
</div>
<div class="specification">
<p>The plane <em>Π</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="_2">
<msub>
<mi></mi>
<mn>2</mn>
</msub>
</math></span> contains the points O, A and B and the plane <em>Π</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="_3">
<msub>
<mi></mi>
<mn>3</mn>
</msub>
</math></span> contains the points O, A and C.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the vector equation of the line (BC).</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>Determine whether or not the lines (OA) and (BC) intersect.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the Cartesian equation of the plane <em>Π</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="_1">
<msub>
<mi></mi>
<mn>1</mn>
</msub>
</math></span>, which passes through C and is perpendicular to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\rm{OA}}} ">
<mover>
<mrow>
<mrow>
<mi mathvariant="normal">O</mi>
<mi mathvariant="normal">A</mi>
</mrow>
</mrow>
<mo>→</mo>
</mover>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the line (BC) lies in the plane <em>Π</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="_1">
<msub>
<mi></mi>
<mn>1</mn>
</msub>
</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>Verify that 2<strong><em>j </em></strong>+ <strong><em>k </em></strong>is perpendicular to the plane <em>Π</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="_2">
<msub>
<mi></mi>
<mn>2</mn>
</msub>
</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>Find a vector perpendicular to the plane <em>Π</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="_3">
<msub>
<mi></mi>
<mn>3</mn>
</msub>
</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>Find the acute angle between the planes <em>Π</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="_2">
<msub>
<mi></mi>
<mn>2</mn>
</msub>
</math></span> and <em>Π</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="_3">
<msub>
<mi></mi>
<mn>3</mn>
</msub>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">g.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the planes <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>Π</mtext><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>Π</mtext><mn>2</mn></msub></math> with the following equations.</p>
<p style="padding-left: 60px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>Π</mtext><mn>1</mn></msub><mtext>: </mtext><mn>3</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>+</mo><mi>z</mi><mo>=</mo><mn>6</mn></math></p>
<p style="padding-left: 60px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>Π</mtext><mn>2</mn></msub><mtext>: </mtext><mi>x</mi><mo>-</mo><mn>2</mn><mi>y</mi><mo>+</mo><mi>z</mi><mo>=</mo><mn>4</mn></math></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find a Cartesian equation of the plane <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>Π</mtext><mn>3</mn></msub></math> which is perpendicular to <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>Π</mtext><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>Π</mtext><mn>2</mn></msub></math> and passes through the origin <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the coordinates of the point where <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>Π</mtext><mn>1</mn></msub></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>Π</mtext><mn>2</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>Π</mtext><mn>3</mn></msub></math> intersect.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Iqbal attempts three practice papers in mathematics. The probability that he passes the first paper is 0.6. Whenever he gains a pass in a paper, his confidence increases so that the probability of him passing the next paper increases by 0.1. Whenever he fails a paper the probability of him passing the next paper is 0.6.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Complete the given probability tree diagram for Iqbal’s three attempts, labelling each branch with the correct probability.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the probability that Iqbal passes at least two of the papers he attempts.</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 probability that Iqbal passes his third paper, given that he passed only one previous paper.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The diagram shows two circles with centres at the points A and B and radii <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2r">
<mn>2</mn>
<mi>r</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span>, respectively. The point B lies on the circle with centre A. The circles intersect at the points C and D.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-02-28_om_17.29.37.png" alt="N16/5/MATHL/HP2/ENG/TZ0/09"></p>
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha ">
<mi>α<!-- α --></mi>
</math></span> be the measure of the angle CAD and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta ">
<mi>θ<!-- θ --></mi>
</math></span> be the measure of the angle CBD in radians.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for the shaded area in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha ">
<mi>α</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta ">
<mi>θ</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha = 4\arcsin \frac{1}{4}">
<mi>α</mi>
<mo>=</mo>
<mn>4</mn>
<mi>arcsin</mi>
<mo></mo>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span> given that the shaded area is equal to 4.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The plane <em>П</em><sub>1</sub> contains the points P(1, 6, −7) , Q(0, 1, 1) and R(2, 0, −4).</p>
</div>
<div class="specification">
<p>The Cartesian equation of the plane <em>П</em><sub>2</sub> is given by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x - 3y - z = 3">
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>3</mn>
<mi>y</mi>
<mo>−<!-- − --></mo>
<mi>z</mi>
<mo>=</mo>
<mn>3</mn>
</math></span>.</p>
</div>
<div class="specification">
<p>The Cartesian equation of the plane <em>П</em><sub>3</sub> is given by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="ax + by + cz = 1">
<mi>a</mi>
<mi>x</mi>
<mo>+</mo>
<mi>b</mi>
<mi>y</mi>
<mo>+</mo>
<mi>c</mi>
<mi>z</mi>
<mo>=</mo>
<mn>1</mn>
</math></span>.</p>
</div>
<div class="specification">
<p>Consider the case that <em>П</em><sub>3</sub> contains <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="L">
<mi>L</mi>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the Cartesian equation of the plane containing P, Q and R.</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <em>П</em><sub>1</sub> and <em>П</em><sub>2</sub> meet in a line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="L">
<mi>L</mi>
</math></span>, verify that the vector equation of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="L">
<mi>L</mi>
</math></span> can be given by <em><strong>r</strong></em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {\begin{array}{*{20}{c}} {\frac{5}{4}} \\ 0 \\ { - \frac{7}{4}} \end{array}} \right) + \lambda \left( {\begin{array}{*{20}{c}} {\frac{1}{2}} \\ 1 \\ { - \frac{5}{2}} \end{array}} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>5</mn>
<mn>4</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mfrac>
<mn>7</mn>
<mn>4</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>λ</mi>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mfrac>
<mn>5</mn>
<mn>2</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <em>П</em><sub>3</sub> is parallel to the line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="L">
<mi>L</mi>
</math></span>, show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a + 2b - 5c = 0">
<mi>a</mi>
<mo>+</mo>
<mn>2</mn>
<mi>b</mi>
<mo>−</mo>
<mn>5</mn>
<mi>c</mi>
<mo>=</mo>
<mn>0</mn>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="5a - 7c = 4">
<mn>5</mn>
<mi>a</mi>
<mo>−</mo>
<mn>7</mn>
<mi>c</mi>
<mo>=</mo>
<mn>4</mn>
</math></span>.</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>Given that <em>П</em><sub>3</sub> is equally inclined to both <em>П</em><sub>1</sub> and <em>П</em><sub>2</sub>, determine two distinct possible Cartesian equations for <em>П</em><sub>3</sub>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">d.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>A scientist conducted a nine-week experiment on two plants, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math>, of the same species. He wanted to determine the effect of using a new plant fertilizer. Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> was given fertilizer regularly, while Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> was not.</p>
<p>The scientist found that the height of Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>,</mo><mo> </mo><msub><mi>h</mi><mi>A</mi></msub><mo> </mo><mtext>cm</mtext></math>, at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> weeks can be modelled by the function <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>h</mi><mi>A</mi></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mi>sin</mi><mo>(</mo><mn>2</mn><mi>t</mi><mo>+</mo><mn>6</mn><mo>)</mo><mo>+</mo><mn>9</mn><mi>t</mi><mo>+</mo><mn>27</mn></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mn>9</mn></math>.</p>
<p>The scientist found that the height of Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>,</mo><mo> </mo><msub><mi>h</mi><mi>B</mi></msub><mo> </mo><mtext>cm</mtext></math>, at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> weeks can be modelled by the function <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>h</mi><mi>B</mi></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mn>8</mn><mi>t</mi><mo>+</mo><mn>32</mn></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mn>9</mn></math>.</p>
</div>
<div class="specification">
<p>Use the scientist’s models to find the initial height of</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> correct to three significant figures.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> when <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>h</mi><mi>A</mi></msub><mfenced><mi>t</mi></mfenced><mo>=</mo><msub><mi>h</mi><mi>B</mi></msub><mfenced><mi>t</mi></mfenced></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>For <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>></mo><mn>6</mn></math>, prove that Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> was always taller than Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>For <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mn>9</mn></math>, find the total amount of time when the rate of growth of Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> was greater than the rate of growth of Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the following diagram.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p>The sides of the equilateral triangle ABC have lengths 1 m. The midpoint of [AB] is denoted by P. The circular arc AB has centre, M, the midpoint of [CP].</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find AM.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}\mathop {\text{M}}\limits^ \wedge {\text{P}}">
<mrow>
<mtext>A</mtext>
</mrow>
<mover>
<mrow>
<mtext>M</mtext>
</mrow>
<mo>∧</mo>
</mover>
<mo></mo>
<mrow>
<mtext>P</mtext>
</mrow>
</math></span> in radians.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area of the shaded region.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = {\text{tan}}\left( {x + \pi } \right){\text{cos}}\left( {x - \frac{\pi }{2}} \right)">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mtext>tan</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mi>π</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>cos</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 < x < \frac{\pi }{2}">
<mn>0</mn>
<mo><</mo>
<mi>x</mi>
<mo><</mo>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</math></span>.</p>
<p>Express <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> in terms of sin <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> and cos <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>.</p>
</div>
<br><hr><br><div class="specification">
<p>Consider the vectors <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">a</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">b</mi></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">a</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>12</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>5</mn></mtd></mtr></mtable></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mi mathvariant="bold-italic">b</mi></mfenced><mo>=</mo><mn>15</mn></math>.</p>
</div>
<div class="specification">
<p>Consider the vector <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">p</mi></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">p</mi><mo>=</mo><mi mathvariant="bold-italic">a</mi><mo>+</mo><mi mathvariant="bold-italic">b</mi></math>.</p>
</div>
<div class="specification">
<p>Consider the vector <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">q</mi></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">q</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>,</mo><mo> </mo><mi>y</mi><mo>∈</mo><msup><mi mathvariant="normal">ℝ</mi><mo>+</mo></msup></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the possible range of values for <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mrow><mi mathvariant="bold-italic">a</mi><mo>+</mo><mi mathvariant="bold-italic">b</mi></mrow></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>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mrow><mi mathvariant="bold-italic">a</mi><mo>+</mo><mi mathvariant="bold-italic">b</mi></mrow></mfenced></math> is a minimum, find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">p</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 <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">q</mi></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>|</mo><mi mathvariant="bold-italic">q</mi><mo>|</mo><mo>=</mo><mo>|</mo><mi mathvariant="bold-italic">b</mi><mo>|</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">q</mi></math> is perpendicular to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">a</mi></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Three points <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mfenced><mrow><mn>3</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn></mrow></mfenced><mo>,</mo><mo> </mo><mtext>B</mtext><mfenced><mrow><mn>0</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>2</mn><mo>,</mo><mo> </mo><mn>0</mn></mrow></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>C</mtext><mfenced><mrow><mn>1</mn><mo>,</mo><mo> </mo><mn>1</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>7</mn></mrow></mfenced></math> lie on the plane <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Π</mi><mn>1</mn></msub></math>.</p>
</div>
<div class="specification">
<p>Plane <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Π</mi><mn>2</mn></msub></math> has equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>x</mi><mo>-</mo><mi>y</mi><mo>+</mo><mn>2</mn><mi>z</mi><mo>=</mo><mn>2</mn></math>.</p>
</div>
<div class="specification">
<p>The plane <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Π</mi><mn>3</mn></msub></math> is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>x</mi><mo>-</mo><mn>2</mn><mi>z</mi><mo>=</mo><mn>3</mn></math>. The line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> and the plane <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Π</mi><mn>3</mn></msub></math> intersect at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>.</p>
</div>
<div class="specification">
<p>The point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>2</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math> lies on <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the vector <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mtext>AB</mtext><mo>→</mo></mover></math> and the vector <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mtext>AC</mtext><mo>→</mo></mover></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the equation of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Π</mi><mn>1</mn></msub></math>, expressing your answer in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi><mi>y</mi><mo>+</mo><mi>c</mi><mi>z</mi><mo>=</mo><mi>d</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>,</mo><mo> </mo><mi>c</mi><mo>,</mo><mo> </mo><mi>d</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> is the intersection of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Π</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Π</mi><mn>2</mn></msub></math>. Verify that the vector equation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> can be written as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>λ</mi><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced></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>Show that at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mo>,</mo><mo> </mo><mi>λ</mi><mo>=</mo><mfrac><mn>3</mn><mn>4</mn></mfrac></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 coordinates of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the reflection of the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> in the plane <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Π</mi><mn>3</mn></msub></math>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the vector equation of the line formed when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> is reflected in the plane <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Π</mi><mn>3</mn></msub></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>The points <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mfenced><mrow><mn>5</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>2</mn><mo>,</mo><mo> </mo><mn>5</mn></mrow></mfenced></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext><mfenced><mrow><mn>5</mn><mo>,</mo><mo> </mo><mn>4</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>1</mn></mrow></mfenced></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>C</mtext><mfenced><mrow><mo>-</mo><mn>1</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>2</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>1</mn></mrow></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>D</mtext><mfenced><mrow><mn>7</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>4</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>3</mn></mrow></mfenced></math> are the vertices of a right-pyramid.</p>
</div>
<div class="specification">
<p>The line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> passes through the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>D</mtext></math> and is perpendicular to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Π</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the vectors <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mtext>AB</mtext><mo>→</mo></mover></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mtext>AC</mtext><mo>→</mo></mover></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use a vector method to show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext><mover><mtext>A</mtext><mo>^</mo></mover><mtext>C</mtext><mo>=</mo><mn>60</mn><mo>°</mo></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the Cartesian equation of the plane <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Π</mi></math> that contains the triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>ABC</mtext></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>z</mi><mo>=</mo><mo>-</mo><mn>2</mn></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find a vector equation of the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</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 determine the minimum distance, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>d</mi><mtext>min</mtext></msub></math>, from <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>D</mtext></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Π</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>Find the volume of right-pyramid <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>ABCD</mtext></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question">
<p>In triangle ABC, AB = 5, BC = 14 and AC = 11.</p>
<p>Find all the interior angles of the triangle. Give your answers in degrees to one decimal place.</p>
</div>
<br><hr><br><div class="specification">
<p>The following shape consists of three arcs of a circle, each with centre at the opposite vertex of an equilateral triangle as shown in the diagram.</p>
<p style="text-align: center;"><img 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"></p>
<p style="text-align: left;">For this shape, calculate</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the perimeter.</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 area.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Barry is at the top of a cliff, standing 80 m above sea level, and observes two yachts in the sea.<br>“<em>Seaview</em>” <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(S)"> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> </math></span> is at an angle of depression of 25°.<br>“<em>Nauti Buoy</em>” <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(N)"> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> </math></span> is at an angle of depression of 35°.<br>The following three dimensional diagram shows Barry and the two yachts at S and N.<br>X lies at the foot of the cliff and angle <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{SXN}} = "> <mrow> <mtext>SXN</mtext> </mrow> <mo>=</mo> </math></span> 70°.</p>
<p><img src="images/Schermafbeelding_2018-02-08_om_11.45.43.png" alt="N17/5/MATHL/HP2/ENG/TZ0/05"></p>
<p>Find, to 3 significant figures, the distance between the two yachts.</p>
</div>
<br><hr><br><div class="specification">
<p>Consider <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mrow><mtext>arctan</mtext><mfenced><mrow><mi>cos</mi><mo> </mo><mi>x</mi></mrow></mfenced><mo>-</mo><mi>k</mi></mrow><msup><mi>x</mi><mn>2</mn></msup></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that a finite limit only exists for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></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>Using l’Hôpital’s rule, show algebraically that the value of the limit is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Given that <strong><em>a</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \times ">
<mo>×</mo>
</math></span> <strong><em>b</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = ">
<mo>=</mo>
</math></span> <strong><em>b</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \times ">
<mo>×</mo>
</math></span> <strong><em>c</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \ne ">
<mo>≠</mo>
</math></span> <strong>0 </strong>prove that <strong><em>a</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" + ">
<mo>+</mo>
</math></span> <strong><em>c</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = ">
<mo>=</mo>
</math></span> <em>s<strong>b </strong></em>where <em>s </em>is a scalar.</p>
</div>
<br><hr><br><div class="question">
<p>Two ships, A and B , are observed from an origin O. Relative to O, their position vectors at time <em>t</em> hours after midday are given by</p>
<p style="padding-left:180px;"><em><strong>r</strong></em><sub>A</sub> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 4 \\ 3 \end{array}} \right) + t\left( {\begin{array}{*{20}{c}} 5 \\ 8 \end{array}} \right)"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>t</mi> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>8</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p style="padding-left:180px;"><em><strong>r</strong></em><sub>B</sub> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 7 \\ { - 3} \end{array}} \right) + t\left( {\begin{array}{*{20}{c}} 0 \\ {12} \end{array}} \right)"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>7</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>t</mi> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>12</mn> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p>where distances are measured in kilometres.</p>
<p>Find the minimum distance between the two ships.</p>
</div>
<br><hr><br><div class="specification">
<p>In a triangle <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ABC, AB}} = 4{\text{ cm, BC}} = 3{\text{ cm}}">
<mrow>
<mtext>ABC, AB</mtext>
</mrow>
<mo>=</mo>
<mn>4</mn>
<mrow>
<mtext> cm, BC</mtext>
</mrow>
<mo>=</mo>
<mn>3</mn>
<mrow>
<mtext> cm</mtext>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\rm{B\hat AC}} = \frac{\pi }{9}">
<mrow>
<mrow>
<mi mathvariant="normal">B</mi>
<mrow>
<mover>
<mi mathvariant="normal">A</mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
<mi mathvariant="normal">C</mi>
</mrow>
</mrow>
<mo>=</mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>9</mn>
</mfrac>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the cosine rule to find the two possible values for AC.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the difference between the areas of the two possible triangles ABC.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Two boats <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> travel due north.</p>
<p>Initially, boat <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> is positioned <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>50</mn></math> metres due east of boat <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>.</p>
<p>The distances travelled by boat <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> and boat <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math>, after <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> seconds, are <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> metres and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> metres respectively. The angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math> is the radian measure of the bearing of boat <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> from boat <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>. This information is shown on the following diagram.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mo>+</mo><mn>50</mn><mo> </mo><mtext>cot</mtext><mo> </mo><mi>θ</mi></math> .</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>At time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>, the following conditions are true.</p>
<p style="padding-left:60px;">Boat <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> has travelled <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> metres further than boat <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>.<br>Boat <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> is travelling at double the speed of boat <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>.<br>The rate of change of the angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>0</mn><mo>.</mo><mn>1</mn></math> radians per second.</p>
<p>Find the speed of boat <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>This diagram shows a metallic pendant made out of four equal sectors of a larger circle of radius <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{OB}} = 9{\text{ cm}}">
<mrow>
<mtext>OB</mtext>
</mrow>
<mo>=</mo>
<mn>9</mn>
<mrow>
<mtext> cm</mtext>
</mrow>
</math></span> and four equal sectors of a smaller circle of radius <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{OA}} = 3{\text{ cm}}">
<mrow>
<mtext>OA</mtext>
</mrow>
<mo>=</mo>
<mn>3</mn>
<mrow>
<mtext> cm</mtext>
</mrow>
</math></span>.<br>The angle <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{BOC}} = ">
<mrow>
<mtext>BOC</mtext>
</mrow>
<mo>=</mo>
</math></span> 20°.</p>
<p><img src="images/Schermafbeelding_2018-02-08_om_11.16.43.png" alt="N17/5/MATHL/HP2/ENG/TZ0/03"></p>
<p>Find the area of the pendant.</p>
</div>
<br><hr><br><div class="question">
<p>Find the Cartesian equation of plane <em>Π</em> containing the points <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}\left( {6,{\text{ }}2,{\text{ }}1} \right)">
<mrow>
<mtext>A</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>6</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>2</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{B}}\left( {3,{\text{ }} - 1,{\text{ }}1} \right)">
<mrow>
<mtext>B</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>3</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo>−</mo>
<mn>1</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and perpendicular to the plane <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x + 2y - z - 6 = 0">
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
<mi>y</mi>
<mo>−</mo>
<mi>z</mi>
<mo>−</mo>
<mn>6</mn>
<mo>=</mo>
<mn>0</mn>
</math></span>.</p>
</div>
<br><hr><br><div class="question">
<p>Find the acute angle between the planes with equations <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x + y + z = 3"> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>+</mo> <mi>z</mi> <mo>=</mo> <mn>3</mn> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2x - z = 2"> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mi>z</mi> <mo>=</mo> <mn>2</mn> </math></span>.</p>
</div>
<br><hr><br><div class="question">
<p>Boat A is situated 10km away from boat B, and each boat has a marine radio transmitter on board. The range of the transmitter on boat A is 7km, and the range of the transmitter on boat B is 5km. The region in which both transmitters can be detected is represented by the shaded region in the following diagram. Find the area of this region.</p>
<p style="text-align: center;"><img 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"></p>
</div>
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