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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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> be the median</p>
<p><strong><br>EITHER</strong></p>
<p>attempts to find the area of the required triangle <em><strong>M1</strong></em></p>
<p>base is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>m</mi><mo>-</mo><mi>a</mi></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p>and height is <math xmlns="http://www.w3.org/1998/Math/MathML"><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>m</mi><mo>-</mo><mi>a</mi></mrow></mfenced></math></p>
<p>area <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mrow><mi>m</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mo>×</mo><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>m</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><msup><mfenced><mrow><mi>m</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mn>2</mn></msup><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></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p>attempts to integrate the correct function <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mi>a</mi><mi>m</mi></munderover><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><mo>d</mo><mi>x</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><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><msubsup><mfenced open="[" close="]"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mn>2</mn></msup></mrow></mfenced><mi>a</mi><mi>m</mi></msubsup></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><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><msubsup><mfenced open="[" close="]"><mrow><mfrac><msup><mi>x</mi><mn>2</mn></msup><mn>2</mn></mfrac><mo>-</mo><mi>a</mi><mi>x</mi></mrow></mfenced><mi>a</mi><mi>m</mi></msubsup></math> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong> </em>for correct integration and <em><strong>A1</strong> </em>for correct limits.</p>
<p> </p>
<p><strong>THEN</strong></p>
<p>sets up (their) <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mi>a</mi><mi>m</mi></munderover><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><mo>d</mo><mi>x</mi></math> or area <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math> <em><strong>M1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>M0A0A0M1A0A0</strong></em> if candidates conclude that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>></mo><mi>c</mi></math> and set up their area or sum of integrals <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math>.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mfenced><mrow><mi>m</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mn>2</mn></msup><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><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mi>a</mi><mo>±</mo><msqrt><mfrac><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><mn>2</mn></mfrac></msqrt></math> <em><strong>(A1)</strong></em></p>
<p> </p>
<p>as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>></mo><mi>a</mi></math>, rejects <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mi>a</mi><mo>-</mo><msqrt><mfrac><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><mn>2</mn></mfrac></msqrt></math></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mi>a</mi><mo>+</mo><msqrt><mfrac><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><mn>2</mn></mfrac></msqrt></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><strong>METHOD 1</strong></p>
<p>number of possible “deals” <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 4! = 24">
<mo>=</mo>
<mn>4</mn>
<mo>!</mo>
<mo>=</mo>
<mn>24</mn>
</math></span> <strong><em>A1</em></strong></p>
<p>consider ways of achieving “no matches” (Chloe winning):</p>
<p>Selena could deal B, C, D (<em>ie</em>, 3 possibilities)</p>
<p>as her first card <strong><em>R1</em></strong></p>
<p>for each of these matches, there are only 3 possible combinations for the remaining 3 cards <strong><em>R1</em></strong></p>
<p>so no. ways achieving no matches <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 3 \times 3 = 9">
<mo>=</mo>
<mn>3</mn>
<mo>×</mo>
<mn>3</mn>
<mo>=</mo>
<mn>9</mn>
</math></span> <strong><em>M1A1</em></strong></p>
<p>so probability Chloe wins <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{9}{{23}} = \frac{3}{8}">
<mo>=</mo>
<mfrac>
<mn>9</mn>
<mrow>
<mn>23</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>3</mn>
<mn>8</mn>
</mfrac>
</math></span> <strong><em>A1AG</em></strong></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>number of possible “deals” <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 4! = 24">
<mo>=</mo>
<mn>4</mn>
<mo>!</mo>
<mo>=</mo>
<mn>24</mn>
</math></span> <strong><em>A1</em></strong></p>
<p>consider ways of achieving a match (Selena winning)</p>
<p>Selena card A can match with Chloe card A<em>, </em>giving 6 possibilities for this happening <strong><em>R1</em></strong></p>
<p>if Selena deals B as her first card, there are only 3 possible combinations for the remaining 3 cards. Similarly for dealing C and dealing D <strong><em>R1</em></strong></p>
<p>so no. ways achieving one match is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 6 + 3 + 3 + 3 = 15">
<mo>=</mo>
<mn>6</mn>
<mo>+</mo>
<mn>3</mn>
<mo>+</mo>
<mn>3</mn>
<mo>+</mo>
<mn>3</mn>
<mo>=</mo>
<mn>15</mn>
</math></span> <strong><em>M1A1</em></strong></p>
<p>so probability Chloe wins <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 1 - \frac{{15}}{{24}} = \frac{3}{8}">
<mo>=</mo>
<mn>1</mn>
<mo>−</mo>
<mfrac>
<mrow>
<mn>15</mn>
</mrow>
<mrow>
<mn>24</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>3</mn>
<mn>8</mn>
</mfrac>
</math></span> <strong><em>A1AG</em></strong></p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p>systematic attempt to find number of outcomes where Chloe wins (no matches)</p>
<p>(using tree diag. or otherwise) <strong><em>M1</em></strong></p>
<p>9 found <strong><em>A1</em></strong></p>
<p>each has probability <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{4} \times \frac{1}{3} \times \frac{1}{2} \times 1">
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>×</mo>
<mn>1</mn>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{{24}}">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>24</mn>
</mrow>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p>their 9 multiplied by their <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{{24}}">
<mfrac>
<mn>1</mn>
<mrow>
<mn>24</mn>
</mrow>
</mfrac>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{3}{8}">
<mo>=</mo>
<mfrac>
<mn>3</mn>
<mn>8</mn>
</mfrac>
</math></span> <strong><em>AG</em></strong></p>
<p> </p>
<p><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X \sim {\text{B}}\left( {50,{\text{ }}\frac{3}{8}} \right)">
<mi>X</mi>
<mo>∼</mo>
<mrow>
<mtext>B</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>50</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mfrac>
<mn>3</mn>
<mn>8</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu = np = 50 \times \frac{3}{8} = \frac{{150}}{8}{\text{ }}\left( { = \frac{{75}}{4}} \right){\text{ }}( = 18.75)">
<mi>μ</mi>
<mo>=</mo>
<mi>n</mi>
<mi>p</mi>
<mo>=</mo>
<mn>50</mn>
<mo>×</mo>
<mfrac>
<mn>3</mn>
<mn>8</mn>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mn>150</mn>
</mrow>
<mn>8</mn>
</mfrac>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>75</mn>
</mrow>
<mn>4</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mo>=</mo>
<mn>18.75</mn>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(M1)A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\sigma ^2} = np(1 - p) = 50 \times \frac{3}{8} \times \frac{5}{8} = \frac{{750}}{{64}}{\text{ }}\left( { = \frac{{375}}{{32}}} \right){\text{ }}( = 11.7)">
<mrow>
<msup>
<mi>σ</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mi>n</mi>
<mi>p</mi>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−</mo>
<mi>p</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>50</mn>
<mo>×</mo>
<mfrac>
<mn>3</mn>
<mn>8</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mn>5</mn>
<mn>8</mn>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mn>750</mn>
</mrow>
<mrow>
<mn>64</mn>
</mrow>
</mfrac>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>375</mn>
</mrow>
<mrow>
<mn>32</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mo>=</mo>
<mn>11.7</mn>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(M1)A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p>mode is 0 <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt at integration by parts <em><strong> (M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}u}}{{{\text{d}}x}} = \frac{1}{{\sqrt {1 - {x^2}} }}{\text{,}}\,\,{\text{d}}v = {\text{d}}x"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>u</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <msqrt> <mn>1</mn> <mo>−</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> </mfrac> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>v</mi> <mo>=</mo> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = x\,{\text{arcsin}}\,x - \int {\frac{{x{\text{d}}x}}{{\sqrt {1 - {x^2}} }}} "> <mo>=</mo> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>arcsin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>−</mo> <mo>∫</mo> <mrow> <mfrac> <mrow> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> <mrow> <msqrt> <mn>1</mn> <mo>−</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> </mfrac> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = x\,{\text{arcsin}}\,x + \sqrt {1 - {x^2}} \left( { + c} \right)"> <mo>=</mo> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>arcsin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <msqrt> <mn>1</mn> <mo>−</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </msqrt> <mrow> <mo>(</mo> <mrow> <mo>+</mo> <mi>c</mi> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int\limits_0^1 {\left( {\pi - {\text{arcsin}}\,x} \right)} \,{\text{d}}x = \left[ {\pi x - x\,{\text{arcsin}}\,x - \sqrt {1 - {x^2}} } \right]_0^1"> <munderover> <mo>∫</mo> <mn>0</mn> <mn>1</mn> </munderover> <mrow> <mrow> <mo>(</mo> <mrow> <mi>π</mi> <mo>−</mo> <mrow> <mtext>arcsin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> <msubsup> <mrow> <mo>[</mo> <mrow> <mi>π</mi> <mi>x</mi> <mo>−</mo> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>arcsin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>−</mo> <msqrt> <mn>1</mn> <mo>−</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> <mo>]</mo> </mrow> <mn>0</mn> <mn>1</mn> </msubsup> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {\pi - \frac{\pi }{2} - 0} \right) - \left( {0 - 0 - 1} \right) = \frac{\pi }{2} + 1"> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>π</mi> <mo>−</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> <mo>−</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mo>−</mo> <mn>0</mn> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> <mo>+</mo> <mn>1</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{\pi + 2}}{2}"> <mo>=</mo> <mfrac> <mrow> <mi>π</mi> <mo>+</mo> <mn>2</mn> </mrow> <mn>2</mn> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int\limits_0^1 k \left( {\pi - {\text{arcsin}}\,x} \right){\text{d}}x = 1"> <munderover> <mo>∫</mo> <mn>0</mn> <mn>1</mn> </munderover> <mi>k</mi> <mrow> <mo>(</mo> <mrow> <mi>π</mi> <mo>−</mo> <mrow> <mtext>arcsin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> <mn>1</mn> </math></span> <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> This line can be seen (or implied) anywhere.</p>
<p><strong>Note: </strong>Do not allow <em><strong>FT A</strong></em> marks from bi to bii.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k\left( {\frac{{\pi + 2}}{2}} \right) = 1"> <mi>k</mi> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mi>π</mi> <mo>+</mo> <mn>2</mn> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>1</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow k = \frac{2}{{2 + \pi }}"> <mo stretchy="false">⇒</mo> <mi>k</mi> <mo>=</mo> <mfrac> <mn>2</mn> <mrow> <mn>2</mn> <mo>+</mo> <mi>π</mi> </mrow> </mfrac> </math></span> <em><strong>AG</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to use product rule to differentiate <em><strong>M1</strong></em></p>
<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 + \frac{{{x^2}}}{{2\sqrt {1 - {x^2}} }} - \frac{1}{{4\sqrt {1 - {x^2}} }} - \frac{{{x^2}}}{{4\sqrt {1 - {x^2}} }} + \frac{{\sqrt {1 - {x^2}} }}{4}"> <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> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>2</mn> <msqrt> <mn>1</mn> <mo>−</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> </mfrac> <mo>−</mo> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <msqrt> <mn>1</mn> <mo>−</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> </mfrac> <mo>−</mo> <mfrac> <mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>4</mn> <msqrt> <mn>1</mn> <mo>−</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msqrt> <mn>1</mn> <mo>−</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> <mn>4</mn> </mfrac> </math></span> <em><strong>A2</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A2</strong></em> for all terms correct, <em><strong>A1</strong></em> for 4 correct terms.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = x\,{\text{arcsin}}\,x + \frac{{2{x^2}}}{{4\sqrt {1 - {x^2}} }} - \frac{1}{{4\sqrt {1 - {x^2}} }} - \frac{{{x^2}}}{{4\sqrt {1 - {x^2}} }} + \frac{{1 - {x^2}}}{{4\sqrt {1 - {x^2}} }}"> <mo>=</mo> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>arcsin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mfrac> <mrow> <mn>2</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>4</mn> <msqrt> <mn>1</mn> <mo>−</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> </mfrac> <mo>−</mo> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <msqrt> <mn>1</mn> <mo>−</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> </mfrac> <mo>−</mo> <mfrac> <mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>4</mn> <msqrt> <mn>1</mn> <mo>−</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>4</mn> <msqrt> <mn>1</mn> <mo>−</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for equivalent combination of correct terms over a common denominator.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = x\,{\text{arcsin}}\,x"> <mo>=</mo> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>arcsin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </math></span> <em><strong>AG</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{E}}\left( X \right) = k\int\limits_0^1 {x\left( {\pi - {\text{arcsin}}\,x} \right)} \,{\text{d}}x"> <mrow> <mtext>E</mtext> </mrow> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>k</mi> <munderover> <mo>∫</mo> <mn>0</mn> <mn>1</mn> </munderover> <mrow> <mi>x</mi> <mrow> <mo>(</mo> <mrow> <mi>π</mi> <mo>−</mo> <mrow> <mtext>arcsin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = k\int\limits_0^1 {\left( {\pi x - x\,{\text{arcsin}}\,x} \right)} \,{\text{d}}x"> <mo>=</mo> <mi>k</mi> <munderover> <mo>∫</mo> <mn>0</mn> <mn>1</mn> </munderover> <mrow> <mrow> <mo>(</mo> <mrow> <mi>π</mi> <mi>x</mi> <mo>−</mo> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>arcsin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = k\left[ {\frac{{\pi {x^2}}}{2} - \frac{{{x^2}}}{2}{\text{arcsin}}\,x + \frac{1}{4}{\text{arcsin}}\,x - \frac{x}{4}\sqrt {1 - {x^2}} } \right]_0^1"> <mo>=</mo> <mi>k</mi> <msubsup> <mrow> <mo>[</mo> <mrow> <mfrac> <mrow> <mi>π</mi> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> <mn>2</mn> </mfrac> <mo>−</mo> <mfrac> <mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> <mn>2</mn> </mfrac> <mrow> <mtext>arcsin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> <mrow> <mtext>arcsin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>−</mo> <mfrac> <mi>x</mi> <mn>4</mn> </mfrac> <msqrt> <mn>1</mn> <mo>−</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> <mo>]</mo> </mrow> <mn>0</mn> <mn>1</mn> </msubsup> </math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for first term, <em><strong>A1</strong></em> for next 3 terms.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = k\left[ {\left( {\frac{\pi }{2} - \frac{\pi }{4} + \frac{\pi }{8}} \right) - \left( 0 \right)} \right]"> <mo>=</mo> <mi>k</mi> <mrow> <mo>[</mo> <mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> <mo>−</mo> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> <mo>+</mo> <mfrac> <mi>π</mi> <mn>8</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {\frac{2}{{2 + \pi }}} \right)\frac{{3\pi }}{8}"> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>2</mn> <mrow> <mn>2</mn> <mo>+</mo> <mi>π</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>8</mn> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{3\pi }}{{4\left( {\pi + 2} \right)}}"> <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> <em><strong>AG</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p>attempt to equate integral to 1 (may appear later) <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k\int\limits_0^6 {\sin \left( {\frac{{\pi x}}{6}} \right){\text{d}}x = 1} "> <mi>k</mi> <munderover> <mo>∫</mo> <mn>0</mn> <mn>6</mn> </munderover> <mrow> <mi>sin</mi> <mo></mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mi>π</mi> <mi>x</mi> </mrow> <mn>6</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></span></p>
<p>correct integral <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k\left[ { - \frac{6}{\pi }\cos \left( {\frac{{\pi x}}{6}} \right)} \right]_0^6 = 1"> <mi>k</mi> <msubsup> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mfrac> <mn>6</mn> <mi>π</mi> </mfrac> <mi>cos</mi> <mo></mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mi>π</mi> <mi>x</mi> </mrow> <mn>6</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mn>0</mn> <mn>6</mn> </msubsup> <mo>=</mo> <mn>1</mn> </math></span></p>
<p>substituting limits <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \frac{6}{\pi }( - 1 - 1) = \frac{1}{k}"> <mo>−</mo> <mfrac> <mn>6</mn> <mi>π</mi> </mfrac> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mfrac> <mn>1</mn> <mi>k</mi> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = \frac{\pi }{{12}}"> <mi>k</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mrow> <mn>12</mn> </mrow> </mfrac> </math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>mean <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 3"> <mo>=</mo> <mn>3</mn> </math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>A1A0A0 </em></strong>for three equal answers in <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(0,{\text{ }}6)"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>6</mn> <mo stretchy="false">)</mo> </math></span>.</p>
<p> </p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>median <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 3"> <mo>=</mo> <mn>3</mn> </math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>A1A0A0 </em></strong>for three equal answers in <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(0,{\text{ }}6)"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>6</mn> <mo stretchy="false">)</mo> </math></span>.</p>
<p> </p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>mode <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 3"> <mo>=</mo> <mn>3</mn> </math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>A1A0A0 </em></strong>for three equal answers in <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(0,{\text{ }}6)"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>6</mn> <mo stretchy="false">)</mo> </math></span>.</p>
<p> </p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{\pi }{{12}}\int\limits_0^2 {\sin } \left( {\frac{{\pi x}}{6}} \right){\text{d}}x"> <mfrac> <mi>π</mi> <mrow> <mn>12</mn> </mrow> </mfrac> <munderover> <mo>∫</mo> <mn>0</mn> <mn>2</mn> </munderover> <mrow> <mi>sin</mi> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mi>π</mi> <mi>x</mi> </mrow> <mn>6</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{\pi }{{12}}\left[ { - \frac{6}{\pi }\cos \left( {\frac{{\pi x}}{6}} \right)} \right]_0^2"> <mo>=</mo> <mfrac> <mi>π</mi> <mrow> <mn>12</mn> </mrow> </mfrac> <msubsup> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mfrac> <mn>6</mn> <mi>π</mi> </mfrac> <mi>cos</mi> <mo></mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mi>π</mi> <mi>x</mi> </mrow> <mn>6</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mn>0</mn> <mn>2</mn> </msubsup> </math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept without the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{\pi }{{12}}"> <mfrac> <mi>π</mi> <mrow> <mn>12</mn> </mrow> </mfrac> </math></span> at this stage if it is added later.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{\pi }{{12}}\left[ { - \frac{6}{\pi }\left( {\cos \frac{\pi }{3} - 1} \right)} \right]"> <mfrac> <mi>π</mi> <mrow> <mn>12</mn> </mrow> </mfrac> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mfrac> <mn>6</mn> <mi>π</mi> </mfrac> <mrow> <mo>(</mo> <mrow> <mi>cos</mi> <mo></mo> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> </math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{4}"> <mo>=</mo> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </math></span> <strong><em>AG</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>from (c)(i) <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{Q_1} = 2"> <mrow> <msub> <mi>Q</mi> <mn>1</mn> </msub> </mrow> <mo>=</mo> <mn>2</mn> </math></span> <strong><em>(A1)</em></strong></p>
<p>as the graph is symmetrical about the middle value <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 3 \Rightarrow {Q_3} = 4"> <mi>x</mi> <mo>=</mo> <mn>3</mn> <mo stretchy="false">⇒</mo> <mrow> <msub> <mi>Q</mi> <mn>3</mn> </msub> </mrow> <mo>=</mo> <mn>4</mn> </math></span> <strong><em>(A1)</em></strong></p>
<p>so interquartile range is</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4 - 2"> <mn>4</mn> <mo>−</mo> <mn>2</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 2"> <mo>=</mo> <mn>2</mn> </math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P(X \leqslant 4|X \geqslant 3) = \frac{{P(3 \leqslant X \leqslant 4)}}{{P(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> <mo>=</mo> <mfrac> <mrow> <mi>P</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo>⩽</mo> <mi>X</mi> <mo>⩽</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>P</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>⩾</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{\frac{1}{4}}}{{\frac{1}{2}}}"> <mo>=</mo> <mfrac> <mrow> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mfrac> </math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{2}"> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p = 1 - \frac{1}{2} - \frac{1}{5} - \frac{1}{5}"> <mi>p</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>−</mo> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> <mo>−</mo> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </math></span> <em><strong> (M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{{10}}"> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mn>10</mn> </mrow> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{E}}\left( X \right)"> <mrow> <mtext>E</mtext> </mrow> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> </math></span> <em><strong> (M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2} + 1 + 2 + \frac{N}{{10}} = 10"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mo>+</mo> <mfrac> <mi>N</mi> <mrow> <mn>10</mn> </mrow> </mfrac> <mo>=</mo> <mn>10</mn> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow N = 65"> <mo stretchy="false">⇒</mo> <mi>N</mi> <mo>=</mo> <mn>65</mn> </math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Do not allow FT in part (b) if their <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span> is outside the range <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 < p < 1"> <mn>0</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mn>1</mn> </math></span>.</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><strong>METHOD 1</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A \cup B) = {\text{P}}(A) + {\text{P}}(B) - {\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> <mi>B</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∩</mo> <mi>B</mi> <mo stretchy="false">)</mo> </math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{P}}(A) + {\text{P}}(A \cap B) + {\text{P}}(A' \cap B) - {\text{P}}(A \cap B)"> <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> <mi>A</mi> <mo>∩</mo> <mi>B</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> <mo>−</mo> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∩</mo> <mi>B</mi> <mo stretchy="false">)</mo> </math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{P}}(A) + {\text{P}}(A' \cap B)"> <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> <strong><em>AG</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A \cup B) = {\text{P}}(A) + {\text{P}}(B) - {\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> <mi>B</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∩</mo> <mi>B</mi> <mo stretchy="false">)</mo> </math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{P}}(A) + {\text{P}}(B) - {\text{P}}(A|B) \times {\text{P}}(B)"> <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> <mi>B</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mrow> <mo stretchy="false">|</mo> </mrow> <mi>B</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{P}}(A) + \left( {1 - {\text{P}}(A|B)} \right) \times {\text{P}}(B)"> <mo>=</mo> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mrow> <mo stretchy="false">|</mo> </mrow> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>×</mo> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{P}}(A) + {\text{P}}(A'|B) \times {\text{P}}(B)"> <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> <mrow> <mo stretchy="false">|</mo> </mrow> <mi>B</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{P}}(A) + {\text{P}}(A' \cap B)"> <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> <strong><em>AG</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(i) use <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> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A' \cap B) = {\text{P}}(B|A'){\text{P}}(A')">
<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>
<mo>=</mo>
<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>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<msup>
<mi>A</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{4}{9} = {\text{P}}(A) + \frac{1}{6}\left( {1 - {\text{P}}(A)} \right)">
<mfrac>
<mn>4</mn>
<mn>9</mn>
</mfrac>
<mo>=</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>6</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="8 = 18{\text{P}}(A) + 3\left( {1 - {\text{P}}(A)} \right)">
<mn>8</mn>
<mo>=</mo>
<mn>18</mn>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mn>3</mn>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>M1</em></strong></p>
<p><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> <strong><em>AG</em></strong></p>
<p>(ii) <strong>METHOD 1</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(B) = {\text{P}}(A \cap B) + {\text{P}}(A' \cap B)">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<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>
<msup>
<mi>A</mi>
<mo>′</mo>
</msup>
<mo>∩</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{P}}(B|A){\text{P}}(A) + {\text{P}}(B|A'){\text{P}}(A')">
<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>
<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>
<mi>B</mi>
<mrow>
<mo stretchy="false">|</mo>
</mrow>
<msup>
<mi>A</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">)</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<msup>
<mi>A</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">)</mo>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{3} \times \frac{1}{3} + \frac{1}{6} \times \frac{2}{3} = \frac{2}{9}">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>6</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>2</mn>
<mn>9</mn>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A \cap B) = {\text{P}}(B|A){\text{P}}(A) \Rightarrow {\text{P}}(A \cap B) = \frac{1}{3} \times \frac{1}{3} = \frac{1}{9}">
<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>B</mi>
<mrow>
<mo stretchy="false">|</mo>
</mrow>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">⇒</mo>
<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>1</mn>
<mn>3</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>9</mn>
</mfrac>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(B) = {\text{P}}(A \cup B) + {\text{P}}(A \cap B) - {\text{P}}(A)">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<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>∩</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>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(B) = \frac{4}{9} + \frac{1}{9} - \frac{1}{3} = \frac{2}{9}">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mn>4</mn>
<mn>9</mn>
</mfrac>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>9</mn>
</mfrac>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>2</mn>
<mn>9</mn>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><img src="images/Schermafbeelding_2017-02-28_om_11.18.41.png" alt="N16/5/MATHL/HP1/ENG/TZ0/02.a/M"> <strong><em>A1A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>A1 </em></strong>for each correct row.</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{E}}(X) = 1 \times \frac{1}{6} + 2 \times \frac{1}{3} + 4 \times \frac{1}{3} + 6 \times \frac{1}{6}">
<mrow>
<mtext>E</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>1</mn>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mn>6</mn>
</mfrac>
<mo>+</mo>
<mn>2</mn>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
<mo>+</mo>
<mn>4</mn>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
<mo>+</mo>
<mn>6</mn>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mn>6</mn>
</mfrac>
</math></span> <strong>(<em>M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{19}}{6}{\text{ }}\left( { = 3\frac{1}{6}} \right)">
<mo>=</mo>
<mfrac>
<mrow>
<mn>19</mn>
</mrow>
<mn>6</mn>
</mfrac>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mn>3</mn>
<mfrac>
<mn>1</mn>
<mn>6</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>If the probabilities in (a) are not values between 0 and 1 or lead to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{E}}(X) > 6">
<mrow>
<mtext>E</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
<mo>></mo>
<mn>6</mn>
</math></span> award <strong><em>M1A0 </em></strong>to correct method using the incorrect probabilities; otherwise allow <strong><em>FT </em></strong>marks.</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p>equating sum of probabilities to 1 (<em>p</em> + 0.5 − <em>p</em> + 0.25 + 0.125 + <em>p</em><sup>3</sup> = 1) <em><strong>M1</strong></em></p>
<p><em>p</em><sup>3</sup> = 0.125 = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{8}"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </math></span></p>
<p><em>p</em>= 0.5 <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><em>μ</em> = 0 × 0.5 + 1 × 0 + 2 × 0.25 + 3 × 0.125 + 4 × 0.125 <em><strong> M1</strong></em></p>
<p>= 1.375 <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( { = \frac{{11}}{8}} \right)"> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mfrac> <mrow> <mn>11</mn> </mrow> <mn>8</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>P(<em>X</em> > <em>μ</em>) = P(<em>X</em> = 2) + P(<em>X</em> = 3) + P(<em>X</em> = 4) <em><strong>(M1)</strong></em></p>
<p>= 0.5 <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Do not award follow through <em><strong>A</strong></em> marks in (b)(i) from an incorrect value of <em>p</em>.</p>
<p><strong>Note:</strong> Award <em><strong>M</strong> </em>marks in both (b)(i) and (b)(ii) provided no negative probabilities, and provided a numerical value for <em>μ</em> has been found.</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><img 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"> <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for a Venn diagram with at least one probability in the correct region.</p>
<p> </p>
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {A \cap B'} \right) = 0.3">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>A</mi>
<mo>∩</mo>
<msup>
<mi>B</mi>
<mo>′</mo>
</msup>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0.3</mn>
</math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {A \cup B} \right) = 0.3 + 0.4 + 0.1 = 0.8">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>A</mi>
<mo>∪</mo>
<mi>B</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0.3</mn>
<mo>+</mo>
<mn>0.4</mn>
<mo>+</mo>
<mn>0.1</mn>
<mo>=</mo>
<mn>0.8</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( B \right) = 0.5">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mi>B</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0.5</mn>
</math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {A \cup B} \right) = 0.5 + 0.4 - 0.1 = 0.8">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>A</mi>
<mo>∪</mo>
<mi>B</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0.5</mn>
<mo>+</mo>
<mn>0.4</mn>
<mo>−</mo>
<mn>0.1</mn>
<mo>=</mo>
<mn>0.8</mn>
</math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( A \right){\text{P}}\left( B \right) = 0.4 \times 0.5">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mi>A</mi>
<mo>)</mo>
</mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mi>B</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0.4</mn>
<mo>×</mo>
<mn>0.5</mn>
</math></span> <em><strong>(M1)</strong></em></p>
<p>= 0.2 <em><strong>A1</strong></em></p>
<p>statement that their <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( A \right){\text{P}}\left( B \right) \ne {\text{P}}\left( {A \cap B} \right)">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mi>A</mi>
<mo>)</mo>
</mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mi>B</mi>
<mo>)</mo>
</mrow>
<mo>≠</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>A</mi>
<mo>∩</mo>
<mi>B</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>R1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>R1</strong> </em>for correct reasoning from their value.</p>
<p>⇒ <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> not independent <strong><em>AG</em></strong></p>
<p> </p>
<p><em><strong>METHOD 2</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {\left. A \right|B} \right) = \frac{{{\text{P}}\left( {A \cap B} \right)}}{{{\text{P}}\left( B \right)}} = \frac{{0.1}}{{0.5}}">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
<mi>A</mi>
<mo>|</mo>
</mrow>
<mi>B</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>A</mi>
<mo>∩</mo>
<mi>B</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mi>B</mi>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mn>0.1</mn>
</mrow>
<mrow>
<mn>0.5</mn>
</mrow>
</mfrac>
</math></span> <em><strong>(M1)</strong></em></p>
<p>= 0.2 <em><strong>A1</strong></em></p>
<p>statement that their <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {\left. A \right|B} \right) \ne {\text{P}}\left( A \right)">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
<mi>A</mi>
<mo>|</mo>
</mrow>
<mi>B</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>≠</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mi>A</mi>
<mo>)</mo>
</mrow>
</math></span> <em><strong>R1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>R1</strong> </em>for correct reasoning from their value.</p>
<p>⇒ <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> not independent <strong><em>AG</em></strong></p>
<p><strong>Note:</strong> Accept equivalent argument using <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {\left. B \right|A} \right) = 0.25">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
<mi>B</mi>
<mo>|</mo>
</mrow>
<mi>A</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0.25</mn>
</math></span>.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>E</mtext><mfenced><mi>X</mi></mfenced><mo>=</mo><mfenced><mrow><mn>0</mn><mo>×</mo><mi>p</mi></mrow></mfenced><mo>+</mo><mfenced><mrow><mn>1</mn><mo>×</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mo>+</mo><mfenced><mrow><mn>2</mn><mo>×</mo><mfrac><mn>1</mn><mn>6</mn></mfrac></mrow></mfenced><mo>+</mo><mn>3</mn><mi>q</mi><mfenced><mrow><mo>=</mo><mfrac><mn>19</mn><mn>12</mn></mfrac></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>⇒</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>+</mo><mn>3</mn><mi>q</mi><mo>=</mo><mfrac><mn>19</mn><mn>12</mn></mfrac></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>6</mn></mfrac><mo>+</mo><mi>q</mi><mo>=</mo><mn>1</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>⇒</mo><mi>p</mi><mo>+</mo><mi>q</mi><mo>=</mo><mfrac><mn>7</mn><mn>12</mn></mfrac></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>attempting integration by parts, <em>eg</em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u = \frac{{\pi x}}{{36}}{\text{,}}\,{\text{d}}u = \frac{\pi }{{36}}{\text{d}}x{\text{,}}\,{\text{d}}v = {\text{sin}}\left( {\frac{{\pi x}}{6}} \right){\text{d}}x{\text{,}}\,v = - \frac{6}{\pi }{\text{cos}}\left( {\frac{{\pi x}}{6}} \right)">
<mi>u</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mi>π</mi>
<mi>x</mi>
</mrow>
<mrow>
<mn>36</mn>
</mrow>
</mfrac>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>u</mi>
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mrow>
<mn>36</mn>
</mrow>
</mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>v</mi>
<mo>=</mo>
<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>d</mtext>
</mrow>
<mi>x</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>v</mi>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mn>6</mn>
<mi>π</mi>
</mfrac>
<mrow>
<mtext>cos</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mi>π</mi>
<mi>x</mi>
</mrow>
<mn>6</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em> (M1)</em></strong></p>
<p>P(0 ≤ <em>X</em> ≤ 3) <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{\pi }{{36}}\left( {\left[ { - \frac{{6x}}{\pi }{\text{cos}}\left( {\frac{{\pi x}}{6}} \right)} \right]_0^3 + \frac{6}{\pi }\int\limits_0^3 {{\text{cos}}\left( {\frac{{\pi x}}{6}} \right)} {\text{d}}x} \right)">
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mrow>
<mn>36</mn>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<msubsup>
<mrow>
<mo>[</mo>
<mrow>
<mo>−</mo>
<mfrac>
<mrow>
<mn>6</mn>
<mi>x</mi>
</mrow>
<mi>π</mi>
</mfrac>
<mrow>
<mtext>cos</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mi>π</mi>
<mi>x</mi>
</mrow>
<mn>6</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>]</mo>
</mrow>
<mn>0</mn>
<mn>3</mn>
</msubsup>
<mo>+</mo>
<mfrac>
<mn>6</mn>
<mi>π</mi>
</mfrac>
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mn>3</mn>
</munderover>
<mrow>
<mrow>
<mtext>cos</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mi>π</mi>
<mi>x</mi>
</mrow>
<mn>6</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> (or equivalent) <strong><em> A1A1</em></strong></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for a correct <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="uv">
<mi>u</mi>
<mi>v</mi>
</math></span> and <em><strong>A1</strong></em> for a correct <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {v\,{\text{d}}u} ">
<mo>∫</mo>
<mrow>
<mi>v</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>u</mi>
</mrow>
</math></span>.</p>
<p>attempting to substitute limits <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{\pi }{{36}}\left[ { - \frac{{6x}}{\pi }{\text{cos}}\left( {\frac{{\pi x}}{6}} \right)} \right]_0^3 = 0">
<mfrac>
<mi>π</mi>
<mrow>
<mn>36</mn>
</mrow>
</mfrac>
<msubsup>
<mrow>
<mo>[</mo>
<mrow>
<mo>−</mo>
<mfrac>
<mrow>
<mn>6</mn>
<mi>x</mi>
</mrow>
<mi>π</mi>
</mfrac>
<mrow>
<mtext>cos</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mi>π</mi>
<mi>x</mi>
</mrow>
<mn>6</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>]</mo>
</mrow>
<mn>0</mn>
<mn>3</mn>
</msubsup>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>(A1)</strong></em></p>
<p>so P(0 ≤ <em>X</em> ≤ 3) <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{\pi }\left[ {{\text{sin}}\left( {\frac{{\pi x}}{6}} \right)} \right]_0^3">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mi>π</mi>
</mfrac>
<msubsup>
<mrow>
<mo>[</mo>
<mrow>
<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>
<mo>]</mo>
</mrow>
<mn>0</mn>
<mn>3</mn>
</msubsup>
</math></span> (or equivalent) <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{\pi }">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mi>π</mi>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[7 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>use of symmetry <em>eg</em> diagram <em><strong>(M1)</strong></em></p>
<p><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</mi>
<mo>+</mo>
<mn>5</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0.2</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<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) = \frac{{{\text{P}}\left( {X > \mu - 5 \cap X < \mu + 5} \right)}}{{{\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>
<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>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo>></mo>
<mi>μ</mi>
<mo>−</mo>
<mn>5</mn>
<mo>∩</mo>
<mi>X</mi>
<mo><</mo>
<mi>μ</mi>
<mo>+</mo>
<mn>5</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<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>
</mrow>
</mfrac>
</math></span> <em><strong>(M1)</strong></em></p>
<p> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{{\text{P}}\left( {\mu - 5 < X < \mu + 5} \right)}}{{{\text{P}}\left( {X > \mu - 5} \right)}}">
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>μ</mi>
<mo>−</mo>
<mn>5</mn>
<mo><</mo>
<mi>X</mi>
<mo><</mo>
<mi>μ</mi>
<mo>+</mo>
<mn>5</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<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>
</mrow>
</mfrac>
</math></span> <em><strong>(A1)</strong></em></p>
<p> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{0.6}}{{0.8}}">
<mo>=</mo>
<mfrac>
<mrow>
<mn>0.6</mn>
</mrow>
<mrow>
<mn>0.8</mn>
</mrow>
</mfrac>
</math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><strong>Note:</strong> <em><strong>A1</strong></em> for denominator is independent of the previous <em><strong>A</strong></em> marks.</p>
<p><strong>OR</strong></p>
<p>use of diagram <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Only award <em><strong>(M1)</strong></em> if the region <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu - 5 < X < \mu + 5">
<mi>μ</mi>
<mo>−</mo>
<mn>5</mn>
<mo><</mo>
<mi>X</mi>
<mo><</mo>
<mi>μ</mi>
<mo>+</mo>
<mn>5</mn>
</math></span> is indicated and used.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {X > \mu - 5} \right) = 0.8">
<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>
<mo>=</mo>
<mn>0.8</mn>
</math></span> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {\mu - 5 < X < \mu + 5} \right) = 0.6">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>μ</mi>
<mo>−</mo>
<mn>5</mn>
<mo><</mo>
<mi>X</mi>
<mo><</mo>
<mi>μ</mi>
<mo>+</mo>
<mn>5</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0.6</mn>
</math></span> <em><strong>(A1)</strong></em></p>
<p><strong>Note:</strong> Probabilities can be shown on the diagram.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{0.6}}{{0.8}}">
<mo>=</mo>
<mfrac>
<mrow>
<mn>0.6</mn>
</mrow>
<mrow>
<mn>0.8</mn>
</mrow>
</mfrac>
</math></span> <em><strong>M1</strong></em><em><strong>A1</strong></em></p>
<p><strong>THEN</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{3}{4}\,\,\, = \left( {0.75} \right)">
<mo>=</mo>
<mfrac>
<mn>3</mn>
<mn>4</mn>
</mfrac>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>0.75</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>attempt to integrate <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>k</mi><msqrt><mn>4</mn><mo>-</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></msqrt></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>k</mi><mfenced open="⌊" close="⌋"><mrow><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac><mtext>arcsin</mtext><mfenced><mrow><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac><mi>x</mi></mrow></mfenced></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)A0</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>arcsin</mtext><mfenced><mrow><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac><mi>x</mi></mrow></mfenced></math>.<br>Condone absence of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> up to this stage.</p>
<p> </p>
<p>equating their integrand to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><msubsup><mfenced open="[" close="]"><mrow><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac><mtext>arcsin</mtext><mfenced><mrow><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac><mi>x</mi></mrow></mfenced></mrow></mfenced><mn>0</mn><mn>1</mn></msubsup><mo>=</mo><mn>1</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mfrac><mrow><mn>3</mn><msqrt><mn>3</mn></msqrt></mrow><mi mathvariant="normal">π</mi></mfrac></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>E</mtext><mo>(</mo><mi>X</mi><mo>)</mo><mo>=</mo><mfrac><mrow><mn>3</mn><msqrt><mn>3</mn></msqrt></mrow><mi mathvariant="normal">π</mi></mfrac><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mfrac><mi>x</mi><msqrt><mn>4</mn><mo>-</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></msqrt></mfrac><mo>d</mo><mi>x</mi></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Condone absence of limits if seen at a later stage.</p>
<p><br><strong>EITHER</strong></p>
<p>attempt to integrate by inspection <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>3</mn><msqrt><mn>3</mn></msqrt></mrow><mi mathvariant="normal">π</mi></mfrac><mo>×</mo><mo>-</mo><mfrac><mn>1</mn><mn>6</mn></mfrac><mo>∫</mo><mo>-</mo><mn>6</mn><mi>x</mi><msup><mfenced><mrow><mn>4</mn><mo>-</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mo>d</mo><mi>x</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>3</mn><msqrt><mn>3</mn></msqrt></mrow><mi mathvariant="normal">π</mi></mfrac><msubsup><mfenced open="[" close="]"><mrow><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><msqrt><mn>4</mn><mo>-</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></msqrt></mrow></mfenced><mn>0</mn><mn>1</mn></msubsup></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Condone the use of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> up to this stage.</p>
<p><br><strong>OR</strong></p>
<p>for example, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo>=</mo><mn>4</mn><mo>-</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>⇒</mo><mfrac><mrow><mo>d</mo><mi>u</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mn>6</mn><mi>x</mi></math></p>
<p><br><strong>Note:</strong> Other substitutions may be used. For example <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo>=</mo><mo>-</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></math>.</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mfrac><msqrt><mn>3</mn></msqrt><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow></mfrac><msubsup><mo>∫</mo><mn>4</mn><mn>1</mn></msubsup><msup><mi>u</mi><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mo>d</mo><mi>u</mi></math> <em><strong>M1</strong></em></p>
<p><strong><br>Note:</strong> Condone absence of limits up to this stage.</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mfrac><msqrt><mn>3</mn></msqrt><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow></mfrac><msubsup><mfenced open="[" close="]"><mrow><mn>2</mn><msqrt><mi>u</mi></msqrt></mrow></mfenced><mn>4</mn><mn>1</mn></msubsup></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Condone the use of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> up to this stage.</p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><msqrt><mn>3</mn></msqrt><mi mathvariant="normal">π</mi></mfrac></math> <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>A0M1A1A0</strong></em> for their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mfenced open="[" close="]"><mrow><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><msqrt><mn>4</mn><mo>-</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></msqrt></mrow></mfenced></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mfenced open="[" close="]"><mrow><mo>-</mo><mn>2</mn><msqrt><mi>u</mi></msqrt></mrow></mfenced></math> for working with incorrect or no limits.</p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Most candidates who attempted part (a) knew that the integrand must be equated to 1 and only a small proportion of these managed to recognize the standard integral involved here. The effect of 3 in 3<em>x<sup>2</sup></em> was missed by many resulting in very few completely correct answers for this part. Part (b) proved to be challenging for vast majority of the candidates and was poorly done in general. Stronger candidates who made good progress in part (a) were often successful in part (b) as well. Most candidates used a substitution, however many struggled to make progress using this approach. Often when using a substitution, the limits were unchanged. If the function was re-written in terms of <em>x</em>, this did not result in an error in the final answer.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = \frac{3}{{16}}">
<mi>a</mi>
<mo>=</mo>
<mfrac>
<mn>3</mn>
<mrow>
<mn>16</mn>
</mrow>
</mfrac>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b = \frac{5}{{16}}">
<mi>b</mi>
<mo>=</mo>
<mfrac>
<mn>5</mn>
<mrow>
<mn>16</mn>
</mrow>
</mfrac>
</math></span> <em><strong>(M1)A1A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for consideration of the possible outcomes when rolling the two dice.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{E}}\left( T \right) = \frac{{1 + 6 + 15 + 28}}{{16}} = \frac{{25}}{8}\left( { = 3.125} \right)">
<mrow>
<mtext>E</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mi>T</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mn>6</mn>
<mo>+</mo>
<mn>15</mn>
<mo>+</mo>
<mn>28</mn>
</mrow>
<mrow>
<mn>16</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mn>25</mn>
</mrow>
<mn>8</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mn>3.125</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(M1)A1</strong></em></p>
<p><strong>Note:</strong> Allow follow through from part (a) even if probabilities do not add up to 1.</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br>