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<h2>HL Paper 1</h2><div class="question">
<p>A continuous random variable <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> has the probability density function</p>
<p style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced open="{" close><mtable><mtr><mtd><mfrac><mn>2</mn><mrow><mfenced><mrow><mi>b</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mfenced><mrow><mi>c</mi><mo>-</mo><mi>a</mi></mrow></mfenced></mrow></mfrac><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mo>,</mo></mtd><mtd><mi>a</mi><mo>≤</mo><mi>x</mi><mo>≤</mo><mi>c</mi></mtd></mtr><mtr><mtd><mfrac><mn>2</mn><mrow><mfenced><mrow><mi>b</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mfenced><mrow><mi>b</mi><mo>-</mo><mi>c</mi></mrow></mfenced></mrow></mfrac><mfenced><mrow><mi>b</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mo>,</mo></mtd><mtd><mi>c</mi><mo><</mo><mi>x</mi><mo>≤</mo><mi>b</mi></mtd></mtr><mtr><mtd><mn>0</mn><mo>,</mo></mtd><mtd><mtext>otherwise</mtext></mtd></mtr></mtable></mfenced></math>.</p>
<p>The following diagram shows the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>≤</mo><mi>x</mi><mo>≤</mo><mi>b</mi></math>.</p>
<p><img style="display:block;margin-left:auto;margin-right:auto;" src="data:image/png;base64,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"></p>
<p>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>≥</mo><mfrac><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mn>2</mn></mfrac></math>, find an expression for the median of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math>.</p>
</div>
<br><hr><br><div class="specification">
<p>Chloe and Selena play a game where each have four cards showing capital letters A, B, C and D.<br>Chloe lays her cards face up on the table in order A, B, C, D as shown in the following diagram.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-02-07_om_14.39.35.png" alt="N17/5/MATHL/HP1/ENG/TZ0/10"></p>
<p>Selena shuffles her cards and lays them face down on the table. She then turns them over one by one to see if her card matches with Chloe’s card directly above.<br>Chloe wins if <strong>no</strong> matches occur; otherwise Selena wins.</p>
</div>
<div class="specification">
<p>Chloe and Selena repeat their game so that they play a total of 50 times.<br>Suppose the discrete random variable <em>X </em>represents the number of times Chloe wins.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the probability that Chloe wins the game is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{3}{8}">
<mfrac>
<mn>3</mn>
<mn>8</mn>
</mfrac>
</math></span>.</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>Determine the mean of <em>X</em>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the variance of <em>X</em>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>The random variable <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span> has probability density function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> given by</p>
<p><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {k\left( {\pi - {\text{arcsin}}\,x} \right)}&{0 \leqslant x \leqslant 1} \\ 0&{{\text{otherwise}}} \end{array}} \right.,\,\,{\text{where }}k{\text{ is a positive constant}}{\text{.}}">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>{</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mi>k</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>π<!-- π --></mi>
<mo>−<!-- − --></mo>
<mrow>
<mtext>arcsin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>0</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</mi>
<mo>⩽<!-- ⩽ --></mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<mrow>
<mtext>otherwise</mtext>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>where </mtext>
</mrow>
<mi>k</mi>
<mrow>
<mtext> is a positive constant</mtext>
</mrow>
<mrow>
<mtext>.</mtext>
</mrow>
</math></span></p>
</div>
<div class="specification">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \left( {\frac{{{x^2}}}{2}} \right){\text{arcsin}}\,x - \left( {\frac{1}{4}} \right){\text{arcsin}}\,x + \left( {\frac{x}{4}} \right)\sqrt {1 - {x^2}} ">
<mi>y</mi>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>arcsin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>arcsin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>x</mi>
<mn>4</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<msqrt>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</math></span>, show that</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the mode of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X"> <mi>X</mi> </math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {{\text{arcsin}}\,x\,{\text{d}}x} "> <mo>∫</mo> <mrow> <mrow> <mtext>arcsin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = \frac{2}{{2 + \pi }}"> <mi>k</mi> <mo>=</mo> <mfrac> <mn>2</mn> <mrow> <mn>2</mn> <mo>+</mo> <mi>π</mi> </mrow> </mfrac> </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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = x\,{\text{arcsin}}\,x"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>arcsin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{E}}\left( X \right) = \frac{{3\pi }}{{4\left( {\pi + 2} \right)}}"> <mrow> <mtext>E</mtext> </mrow> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mrow> <mn>4</mn> <mrow> <mo>(</mo> <mrow> <mi>π</mi> <mo>+</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>The continuous random variable <em>X</em> has a probability density function given by</p>
<p style="padding-left: 120px;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = \left\{ {\begin{array}{*{20}{l}} {k\sin \left( {\frac{{\pi x}}{6}} \right),}&{0 \leqslant x \leqslant \,6} \\ {0,}&{{\text{otherwise}}} \end{array}} \right.">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<mo>{</mo>
<mrow>
<mtable columnalign="left" rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mi>k</mi>
<mi>sin</mi>
<mo><!-- --></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mi>π<!-- π --></mi>
<mi>x</mi>
</mrow>
<mn>6</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>0</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</mi>
<mo>⩽<!-- ⩽ --></mo>
<mspace width="thinmathspace"></mspace>
<mn>6</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>0</mn>
<mo>,</mo>
</mrow>
</mtd>
<mtd>
<mrow>
<mrow>
<mtext>otherwise</mtext>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k"> <mi>k</mi> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By considering the graph of <em>f </em>write down the mean of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X"> <mi>X</mi> </math></span>;</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By considering the graph of <em>f </em>write down the median of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X"> <mi>X</mi> </math></span>;</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By considering the graph of <em>f </em>write down the mode of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X"> <mi>X</mi> </math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P(0 \leqslant X \leqslant 2) = \frac{1}{4}"> <mi>P</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>⩽</mo> <mi>X</mi> <mo>⩽</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence state the interquartile range 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">c.ii.</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="P(X \leqslant 4|X \geqslant 3)"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>⩽</mo> <mn>4</mn> <mrow> <mo stretchy="false">|</mo> </mrow> <mi>X</mi> <mo>⩾</mo> <mn>3</mn> <mo stretchy="false">)</mo> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>The probability distribution of a discrete random variable, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span>, is given by the following table, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="N">
<mi>N</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span> are constants.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{E}}\left( X \right) = 10"> <mrow> <mtext>E</mtext> </mrow> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>10</mn> </math></span>, find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="N"> <mi>N</mi> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider two events <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A">
<mi>A</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A">
<mi>A</mi>
</math></span> defined in the same sample space.</p>
</div>
<div class="specification">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A \cup B) = \frac{4}{9},{\text{ P}}(B|A) = \frac{1}{3}">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo>∪<!-- ∪ --></mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mn>4</mn>
<mn>9</mn>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mrow>
<mo stretchy="false">|</mo>
</mrow>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(B|A') = \frac{1}{6}">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mrow>
<mo stretchy="false">|</mo>
</mrow>
<msup>
<mi>A</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>6</mn>
</mfrac>
</math></span>,</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A \cup B) = {\text{P}}(A) + {\text{P}}(A' \cap B)"> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∪</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <msup> <mi>A</mi> <mo>′</mo> </msup> <mo>∩</mo> <mi>B</mi> <mo stretchy="false">)</mo> </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>(i) show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A) = \frac{1}{3}">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</math></span>;</p>
<p>(ii) hence find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(B)">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
</math></span>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The faces of a fair six-sided die are numbered 1, 2, 2, 4, 4, 6. Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span> be the discrete random variable that models the score obtained when this die is rolled.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Complete the probability distribution table for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span>.</p>
<p><img src="images/Schermafbeelding_2017-02-28_om_11.16.45.png" alt="N16/5/MATHL/HP1/ENG/TZ0/02.a"></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the expected value 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">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The discrete random variable <em>X</em> has the following probability distribution, where<em> p</em> is a constant.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <em>p</em>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <em>μ</em>, the expected value of <em>X</em>.</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>Find P(<em>X</em> > <em>μ</em>).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider two events, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A">
<mi>A</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="B">
<mi>B</mi>
</math></span>, such that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( A \right) = {\text{P}}\left( {A' \cap B} \right) = 0.4">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mi>A</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<msup>
<mi>A</mi>
<mo>′</mo>
</msup>
<mo>∩<!-- ∩ --></mo>
<mi>B</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0.4</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {A \cap B} \right) = 0.1">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>A</mi>
<mo>∩<!-- ∩ --></mo>
<mi>B</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0.1</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By drawing a Venn diagram, or otherwise, find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {A \cup B} \right)">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>A</mi>
<mo>∪</mo>
<mi>B</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the events <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A">
<mi>A</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="B">
<mi>B</mi>
</math></span> are not independent.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>A discrete random variable <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> has the probability distribution given by the following table.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>E</mtext><mfenced><mi>X</mi></mfenced><mo>=</mo><mfrac><mn>19</mn><mn>12</mn></mfrac></math>, determine the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> and the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math>.</p>
</div>
<br><hr><br><div class="question">
<p>A continuous random variable <em>X</em> has the probability density function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> given by</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {\frac{{\pi x}}{{36}}{\text{sin}}\left( {\frac{{\pi x}}{6}} \right){\text{,}}}&{0 \leqslant x \leqslant 6} \\ {0{\text{,}}}&{{\text{otherwise}}} \end{array}} \right.">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>{</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mi>π</mi>
<mi>x</mi>
</mrow>
<mrow>
<mn>36</mn>
</mrow>
</mfrac>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mi>π</mi>
<mi>x</mi>
</mrow>
<mn>6</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>,</mtext>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>0</mn>
<mo>⩽</mo>
<mi>x</mi>
<mo>⩽</mo>
<mn>6</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>0</mn>
<mrow>
<mtext>,</mtext>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow>
<mrow>
<mtext>otherwise</mtext>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</math></span>.</p>
<p style="text-align: left;">Find P(0 ≤ <em>X</em> ≤ 3).</p>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span> be a random variable which follows a normal distribution with mean <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu ">
<mi>μ<!-- μ --></mi>
</math></span>. Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {X < \mu - 5} \right) = 0.2">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo><</mo>
<mi>μ<!-- μ --></mi>
<mo>−<!-- − --></mo>
<mn>5</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0.2</mn>
</math></span> , find</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {X > \mu + 5} \right)">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo>></mo>
<mi>μ</mi>
<mo>+</mo>
<mn>5</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {X < \mu + 5\,\left| {\,X > \mu - 5} \right.} \right)">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo><</mo>
<mi>μ</mi>
<mo>+</mo>
<mn>5</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>|</mo>
<mrow>
<mspace width="thinmathspace"></mspace>
<mi>X</mi>
<mo>></mo>
<mi>μ</mi>
<mo>−</mo>
<mn>5</mn>
</mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The continuous random variable <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> has probability density function</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced open="{" close><mtable columnspacing="1.4ex" columnalign="left"><mtr><mtd><mfrac><mi>k</mi><msqrt><mn>4</mn><mo>-</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></msqrt></mfrac><mo>,</mo></mtd><mtd><mn>0</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>1</mn></mtd></mtr><mtr><mtd><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mn>0</mn><mo>,</mo></mtd><mtd><mtext>otherwise</mtext><mo>.</mo></mtd></mtr></mtable></mfenced></math></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></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 <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>E</mtext><mo>(</mo><mi>X</mi><mo>)</mo></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Two unbiased tetrahedral (four-sided) dice with faces labelled 1, 2, 3, 4 are thrown and the scores recorded. Let the random variable <em>T</em> be the maximum of these two scores.</p>
<p>The probability distribution of <em>T</em> is given in the following table.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <em>a</em> and the value of <em>b</em>.</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 expected value of <em>T</em>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br>