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<h2>HL Paper 2</h2><div class="specification">
<p>A continuous random variable <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> has a probability density function given by</p>
<p style="padding-left: 180px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced open="{" close><mtable><mtr><mtd><mtext>arccos</mtext><mo> </mo><mi>x</mi><mo> </mo></mtd><mtd><mn>0</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mtext>otherwise</mtext></mtd></mtr></mtable></mfenced></math></p>
<p>The median of this distribution is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mfenced><mrow><mfenced open="|" close="|"><mrow><mi>X</mi><mo>-</mo><mi>m</mi></mrow></mfenced><mo>≤</mo><mi>a</mi></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>3</mn></math>, determine the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A 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</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}} {3a}&{\text{,}}&{0 \leqslant x < 2}&{} \\ {a\left( {x - 5} \right)\left( {1 - x} \right)}&{\text{,}}&{2 \leqslant x \leqslant b}&{a{\text{, }}b \in {\mathbb{R}^ + }{\text{, }}3 < b \leqslant 5.} \\ 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>
<mn>3</mn>
<mi>a</mi>
</mrow>
</mtd>
<mtd>
<mrow>
<mtext>,</mtext>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>0</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</mi>
<mo><</mo>
<mn>2</mn>
</mrow>
</mtd>
<mtd>
<mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mi>a</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>5</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow>
<mtext>,</mtext>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>2</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</mi>
<mo>⩽<!-- ⩽ --></mo>
<mi>b</mi>
</mrow>
</mtd>
<mtd>
<mrow>
<mi>a</mi>
<mrow>
<mtext>, </mtext>
</mrow>
<mi>b</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
<mrow>
<mtext>, </mtext>
</mrow>
<mn>3</mn>
<mo><</mo>
<mi>b</mi>
<mo>⩽<!-- ⩽ --></mo>
<mn>5.</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<mtext>,</mtext>
</mrow>
</mtd>
<mtd>
<mrow>
<mrow>
<mtext>otherwise</mtext>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</math></span></p>
<p> </p>
</div>
<div class="specification">
<p>Consider the case where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b = 5">
<mi>b</mi>
<mo>=</mo>
<mn>5</mn>
</math></span>.</p>
</div>
<div class="specification">
<p>Find the value of</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find, in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span>, the probability that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X"> <mi>X</mi> </math></span> lies between 1 and 3.</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>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span>. State the coordinates of the end points and any local maximum or minimum points, giving your answers in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</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)"> <mrow> <mtext>E</mtext> </mrow> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>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">[4]</div>
<div class="question_part_label">c.iii.</div>
</div>
<br><hr><br><div class="specification">
<p>A continuous random variable <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> has the probability density function <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>n</mi></msub></math> given by</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>n</mi></msub><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced open="{" close><mrow><mtable columnalign="left"><mtr><mtd><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><msup><mi>x</mi><mi>n</mi></msup><mo>,</mo></mtd></mtr><mtr><mtd><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mn>0</mn><mo>,</mo></mtd></mtr></mtable><mo> </mo><mo> </mo></mrow></mfenced><mtable><mtr><mtd><mn>0</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>1</mn></mtd></mtr><mtr><mtd><mtext>otherwise</mtext></mtd></mtr></mtable></math></p>
<p style="text-align: left;">where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>n</mi><mo>≥</mo><mn>0</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>E</mtext><mfenced><mi>X</mi></mfenced><mo>=</mo><mfrac><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></mfrac></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Var</mtext><mfenced><mi>X</mi></mfenced><mo>=</mo><mfrac><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mrow><msup><mfenced><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mn>2</mn></msup><mfenced><mrow><mi>n</mi><mo>+</mo><mn>3</mn></mrow></mfenced></mrow></mfrac></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The times taken for male runners to complete a marathon can be modelled by a normal distribution with a mean 196 minutes and a standard deviation 24 minutes.</p>
</div>
<div class="specification">
<p>It is found that 5% of the male runners complete the marathon in less than <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{T_1}">
<mrow>
<msub>
<mi>T</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> minutes.</p>
</div>
<div class="specification">
<p>The times taken for female runners to complete the marathon can be modelled by a normal distribution with a mean 210 minutes. It is found that 58% of female runners complete the marathon between 185 and 235 minutes.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that a runner selected at random will complete the marathon in less than 3 hours.</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>Calculate <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{T_1}">
<mrow>
<msub>
<mi>T</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the standard deviation of the times taken by female runners.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>A Chocolate Shop advertises free gifts to customers that collect three vouchers. The vouchers are placed at random into 10% of all chocolate bars sold at this shop. Kati buys some of these bars and she opens them one at a time to see if they contain a voucher. Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(X = n)">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>=</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</math></span> be the probability that Kati obtains her third voucher on the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n{\text{th}}">
<mi>n</mi>
<mrow>
<mtext>th</mtext>
</mrow>
</math></span> bar opened.</p>
<p>(It is assumed that the probability that a chocolate bar contains a voucher stays at 10% throughout the question.)</p>
</div>
<div class="specification">
<p>It is given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(X = n) = \frac{{{n^2} + an + b}}{{2000}} \times {0.9^{n - 3}}">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>=</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>n</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>a</mi>
<mi>n</mi>
<mo>+</mo>
<mi>b</mi>
</mrow>
<mrow>
<mn>2000</mn>
</mrow>
</mfrac>
<mo>×<!-- × --></mo>
<mrow>
<msup>
<mn>0.9</mn>
<mrow>
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>3</mn>
</mrow>
</msup>
</mrow>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \geqslant 3,{\text{ }}n \in \mathbb{N}">
<mi>n</mi>
<mo>⩾<!-- ⩾ --></mo>
<mn>3</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>n</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">N</mi>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>Kati’s mother goes to the shop and buys <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> chocolate bars. She takes the bars home for Kati to open.</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}}(X = 3) = 0.001"> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>=</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.001</mn> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(X = 4) = 0.0027"> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>=</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.0027</mn> </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>Find the values of the constants <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>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Deduce that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{P}}(X = n)}}{{{\text{P}}(X = n - 1)}} = \frac{{0.9(n - 1)}}{{n - 3}}"> <mfrac> <mrow> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>=</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>=</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>0.9</mn> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>n</mi> <mo>−</mo> <mn>3</mn> </mrow> </mfrac> </math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n > 3"> <mi>n</mi> <mo>></mo> <mn>3</mn> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>(i) Hence show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X"> <mi>X</mi> </math></span> has two modes <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{m_1}"> <mrow> <msub> <mi>m</mi> <mn>1</mn> </msub> </mrow> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{m_2}"> <mrow> <msub> <mi>m</mi> <mn>2</mn> </msub> </mrow> </math></span>.</p>
<p>(ii) State the values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{m_1}"> <mrow> <msub> <mi>m</mi> <mn>1</mn> </msub> </mrow> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{m_2}"> <mrow> <msub> <mi>m</mi> <mn>2</mn> </msub> </mrow> </math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the minimum value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> such that the probability Kati receives at least one free gift is greater than 0.5.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>Iqbal attempts three practice papers in mathematics. The probability that he passes the first paper is 0.6. Whenever he gains a pass in a paper, his confidence increases so that the probability of him passing the next paper increases by 0.1. Whenever he fails a paper the probability of him passing the next paper is 0.6.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Complete the given probability tree diagram for Iqbal’s three attempts, labelling each branch with the correct probability.</p>
<p style="text-align: center;"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAYEAAAFPCAYAAACru3LgAAAgAElEQVR4Ae2dD2xU153vv9CoL6CIfUvy1B2eo1Z1Y+hTd0P8h337MMLPJJOEJTI18luFCoPAqCuKq01UmxBnd/PUOA0ZhJS1y+pFGNmm7f7xG8tRUwhu7JqH2beBwabb6MHQpMquwW6XUK2BNUnAPk+/wXc8f+7M3Hvnzsy9c79Xsub+OX8/5/j8zjm/c85vkVJKgRcJkAAJkIAnCSz2ZK6ZaRIgARIggQgBCgFWBBIgARLwMAEKAQ8XPrNOAiRAAhQCrAMkQAIk4GECFAIeLnxmnQRIgAQoBFgHSIAESMDDBCgEPFz4zDoJkIAbCUygf/tXsGjRouS/FdtxsP88puaM54tCwDgruiQBEiABBxB4GPU9/4ih1goANQiEbkK2e6nZSbzXAhza8gy2HjqLWwZTSiFgEBSdkQAJkIDjCPj+COWPLL2XrMU+rGn+Np4vncLIoQGcvWFsOEAh4LhSZYJIgARIIAOBu/+MseB5+LY9jsplMc34zDSuzWTwm/A5xnfCFz6SAAmQAAk4ksDcr36On37ow++vXIEHoin8DFff7cexKSQLh6ib5BsKgWQmfEMCJEACDibwCT4YfQeDWIHVX3oIkUZ8bgrn+/8KL+79PqZqXsYP/qwaywzmgELAICg6IwESIAFnELiFK+FfAb6VwE++cW+F0OdWoLLjGtb/dQiTQ3+JWt/nDSd1EU8RNcyKDkmABEggbwRu376NsbExjI+PY+/evQvx3hjGvlUbcGzbEC4dqDXc418IIP7uvvhHPpEACZAACRSSwOXLlzE4OIje3l6cO3cukpTq6mqsXr06cn/3l2MITiXqA6ynmELAOjv6JAESIAFbCPz2t7/FyMhIpOF/6623omG2tLSgrq4OK1eunH/3CX7187P4ENV4sfpL9/QBUdfWbigErHGjLxIgARLImsCZM2dw6tQptLW1RcOSRr+xsRE1NTVYvnx59L3czF09jgMv9QGlATz65fvjvll9oE7AKjn6IwESIAELBK5cuYLh4WF0dnZGp3skmI6ODsRO+8QHfQvnDz6DypaRmNcN6Ar3YmdZdsKAQiAGKW9JgARIIBcENCVvd3c3jhw5Eo2iqakJDQ0NWLduHZYsWRJ9n88bTgflkzbjIgES8BQBTcnb3NwczXdVVVVkumfz5s0oKSmJvi/UDYVAocgzXhIggaIkkEnJu3btWkflm0LAUcXBxJAACbiVgFklr1PySSHglJJgOkiABFxHwJqS11nZpBBwVnkwNSRAAg4n4GQlrxV0FAJWqNEPCZCA5whcuHABo6OjcLKS10qhUAhYoUY/JEACniCQSsnb3t6O9evXw2lKXiuFQiFghRr9kAAJFDUBUfLK8Q2BQCCaz3Q7eaOOXHhDIeDCQmOSSYAE7CcgSt6BgYG4g9skFtnJ6/f7UVZWZn+kDgiRQsABhcAkkAAJFIaAKHlPnz6Nvr6+pJ28O3bsQHl5ecF28uaLCIVAvkgzHhIgAccQSKXklXP7a2trHbGTN1+wKATyRZrxkAAJFJSAF5S8VgBTCFihRj8kQAKuIeAlJa+VQqEQsEKNfkiABBxNwKtKXiuFQiFghRr9kAAJOI4AlbzWioRCwBo3+iIBEnAIAVHyHj9+PM46lxzX7EUlr5UioRCwQo1+SIAECkpAlLxvv/02+vv7I5u6tMTITt6NGzdGjbJr7/mbmgCFQGo2/EICJOAgAtrBbXo7effs2YPKysokm7wOSr5jk0Ih4NiiYcJIgASEgJ6SV7POVcw7efNV+hQC+SLNeEiABAwT0JS8hw8fjpvuEZu8XtnJaxhWlg4pBLIESO8kQAL2EUin5N20aROne+xDHQ2JQiCKgjckQAKFIEAlbyGoL8RJIbDAgnckQAJ5IkAlb55AG4iGQsAAJDohARKwhwCVvPZwtDMUCgE7aTIsEiCBJAJU8iYhcdQLCgFHFQcTQwLFQ0BPySvWuerr60Elr3PKmULAOWXBlJBAgQhMoH/7f8eW3g+T4/c1ItD5bXxjcwV8i5M/p3qzZs0anDt3LvqZO3mjKBx3Y6JYHZd2JogESMAWAg+jvucfMdRaAaAGgdBNKKWgZifxXgtwaMsz2HroLG6ZiOuFF16A9PpPnjyJmZkZvPjiizzKwQS/fDpdpKS0eZEACXicwMcY3vcUNhzzY+jSK6hdNt8/vHseB1dVomVmf/x7j9MqpuxzJFBMpcm8kIBVAnf/GWPB8/BtexyVmgCQsGamcW3GaqD05wYCFAJuKCWmkQRyTGDuVz/HTz/04fdXrsAD0bg+w9V3+3FsCsnCIeqGN24nQCHg9hJk+kkgawKf4IPRdzCIFVj9pYcQaRTmpnC+/6/w4t7vY6rmZfzgz6qxLOt4GIATCVAIOLFUmCYSyIKA2NRtbW2N/BkL5hauhH8F+FYCP/kGFi1ahEWfW4HKjmtY/9chTA79JWp9nzcW1LwrOed/8+bNGBwchOwT4OVcAlQMO7dsmDISMExAbyeueJ6YmEBJSUn6cG4MY9+qDTi2bQiXDtTa0uPnEtH0yJ30lSMBJ5UG00ICJghID1t62rt378bDDz+M5ubmyNp8OW55dHQ0sjQzowAAcPeXYwhOJeoDTCREx+nZs2cxPj4O2R8gV1tbGx577LHI6KC3txdyaBwvZxDgSMAZ5cBUkIBhArITVxp5afS1y7pN3U9w+WgjVu4CusK92Fl2vxakbb/S4IdCISTaBmhpaYnsJSgvL8eSJUtsi48BmSNAIWCOF12TQEEISEM6MjIC6UWLeUXtkp72+vXrsXbtWu2Vqd+5q/3YXbUFR5cGELr0HVTk+AyBy5cvR0Yvkg9tR7FmJUx0CEZGLqYySMeZCchmMV4kQALOJDA6OqpaWlpkQ2f0r66uTgWDQXX9+vUsEn1ThQI10TDvhd+gusK3swjTuNeZmRkleWtqaopLg+Tt5MmTSr7zyg8BjgQyy0m6IIG8Ekil5O3o6EAx2tTVjMp0dnZGRwcCnOcN5afaUQjkhzNjIYG0BLTjlvv6+nDkyJGoW6/Z1OXJo9Giz9sNhUDeUDMiEkgmoNfoWVfyJofv1jdUJuev5CgE8seaMZFAhIA2/SEbqhKVvBs3buRpmwn1JJ0yuRinxxKyn/NHCoGcI2YEJIDIrtmxsbFIox8IBKJI5LjlPXv2oLKyEsuXL4++500yAZkyI8NkLtm+oRDIliD9k0AaAnpKXm1JJHuxacBl+MTRVAZAJj5TCJiARackYISApuRN3BzlNSWvEVZ2uEmnV6EZy8yEKQQyM6ILEjBEgI2RIUw5c0Thaw0thYA1bvRFAhECnJZwZkXgNJzxcqEQMM6KLkkgQoAKSvdUBJZV5rKiEMjMiC5IIEKAvUt3VwSO2vTLj0JAnwvfkkCEgDQceidgUsnr7gpC/c1C+VEILLDgHQlECeg1ErKmv76+HlxxEsXk+hsqkwEKAddXY2bALgLadAEPMrOLqLvC0ZvukxwU68F9WulQCGgk+OtJAprisLu7O+7gNm0n77p162jwxIM1Q+w0y5Eeibu7GxsbUVNTU1S7uykEPFjBmWUg3Xk0NG7CGqIRkNFhLoz5aOE74ZdCwAmlwDTkhUAqJS/NHOYFv+sjET2RfWY9s8Bx9zwOrqpEy4fpw/C1DmHyQG16R/I1P7ZrGAsJFI7A+Pi4am9vT7Jg1dPTk6V1rsLliTEXjoBYPRPrZ4lW0eRZrKXlxyraNTXUWqGAb6rg5J14GDd/oboa/ap16Fr8+xRPHAlklpN04UICVPK6sNBcmOSCKZPnLuHo07XYhVcQPrETZYtj4X2Cyz3HcOXru1C7LO5DrKOF+xTCga9JwHUEaLfWdUVWVAnOnT1oHUzTQ6rV51P+rotqVvs8PaT+/FBIJYwLtK8pfzkSWJCHvHMpASp5XVpwRZrs3CuT53Bj+CWs2vABXgn3YmfZ/cCtS+j/7nfx3pNv4EDtQ6bIUgiYwkXHTiGQ6h+NSl6nlBDTIQRyo0z+BJePNmLlrr4EyA3o0oRCwpd0jxQC6ejwm+MIyPrtU6dOoa2tLZo2WdNfjOu3oxnkjesJaDuT+/r64vajyPEjDQ0NMLUfRUcfMDf1U7y0+31sGXgOFfeZw0UhYI4XXReAgCjfhoeHkbiTV3ZyVldX0yZvAcqEUVonkHV9nurH9hVbcCYQwqXvVCDS5s9dQs8bv8bXn6vBMpNJoxAwCYzO80Mg1U5eSz2n/CSZsZCAaQLmR7aaPmAQ24beMT3/r5dACgE9KnxXMAKakre5uTmaBs0mL3fyRpHwpsgIZNJxrV27dj7HH2N431PY8HolgpOdqPeZnPvR4UYhoAOFrwpDQP4RHnzwwWjkmpJ34R8g+ok3JFC0BPQ6QrJTWf4P5q72Y3fVFhx9IojJnnr4bKBAIWADRAZhH4HBwUHcunWr6A7pso8QQ/IKAW1KdHx8HHv37sD5g8+gsmUkJvvftGU0QCEQg5S3JEACJOA1Agb2FHsNCfNLAiRAAt4hQCHgnbJmTkmABEggiQCFQBISvigkAVEOy1woLxIggfwQoBDID2dvxCKbWBYtwiKdvxXbD6L//BTmMpDYt28fli5dGtkYJqskeJEACeSWAIVAbvl6K3RfPXqmh9Aq69ZKAwjdUWKvArOTo2hBD7ZU7sah8/+WkomMAn73d3838l32CaxcuRKyN6C/vx/yjRcJkIAQmED/9q/odrYWrdiOg/3nMZWptxUDkkIgBgZv7SPg21KOR+b3sSz2rUXzt7ejFD/Bob8fw40U0Sxfvhyvv/46rl+/jmAwCDkTSOy8btmyJbJ/oLW1FbLDkhcJeIWATI3KMRPx18Oo7/lHDLVWAKhBIHQz0tlSs5N4rwU4tOUZbD10FrfiPaV8ohBIiYYfrBC4+8sxBKcqsO3JP4g5w2QOM9O/xYzBAEUY1NfXY2BgAOFwGHJGkFxi9FvOClqzZk1kuij5n8NgBHRGAg4nIKePyllZMjX68MMP6wiC+Qz4/gjljyy997DYhzXN38bzpVMYOTSAszeMDQcoBBxeGdyVvE/wq5+fxYf4MlaWPLCQ9LkJvPvDH2MKicJhwUmqu7KyMuzduxczMzMR+65ydtC5c+cg00Xyz7F7927IBjMqk1MR5Hu3EJApT5n6lCnQxx57LFLHJe3t7e2R+h+Xj7v/jLHgefi2PY7KWOthM9O4ZrS3pQWY0twMP5CAWQKzF1WX36fg26+Gpu/ZO5qdDKlgoFH54FM1+wfVZNQMktnAF9xPTEwosQ9cVVUVZze4o6NDiT1hXiTgJgJWLJLNhruUHwmWxdSn6krwW8oHKF/rkJo2CIGG5g2CojMDBCIm76B8jc+r52t88w20T9W0vqkGQpMLZvAMBGXUifwD6RmRDwaDNCJvFCLd5Z2AdGSk06LXkQmHwxnSc1uFuxoUULFgTH52UoWCAdXog0LNy2po8tMMYSx8phBYYMG7rAjMqumh/coXWzGzCs+c5+vXrytp+Ovq6uJGBy0tLUoEBS8SKDQBsYF98uRJ1dTUFFdH5VnqqHw3dl1TQ60VCr6tqvV5/0JYNa2qayBkerRNIWCMOl1lJHBThQI1CmhQXeHbGV3n0oH0pKSXBSD6Jz0ueSc9MF4kkE8CMkWZOFqV+ihTmpbqozbiNjHlky6/FALp6PCbcQKaPsDfpcI2zPsbjzi1S+lZSQ9Lr+clPTLjPa/UcfALCegRkJGpNPKJI1MRBtnqre6EAqo0SR+glwpj7ygEjHHypCvjjeSCQqo0EFJ3HEiLymQHFkqRJUnrdMgUZOwoVASBdDpEMGR/afoA+0bcFALZl0rRhSCVVVNaZRQEd0IqULow7RKp/A4aDegVTiplsvTc7PlH1YuV74qVgJ6SV5t+zKzkNUdl9kpQ7RTlb2lAhWzqbVEImCuDonatNf6xvRjpwRTrJfmV/CUO2TVlckYBWKxgmK+MBKRu6NUd80rejFHNO9B0brEdLntGAzQqo22Y8PCvbFL50Y9+FN2cIijEru8LL7yAp59+GkuWLCl6OppJv97e3shmNI1BY2NjZPNOSUlJ0TNgBjMTkJ28x48fR1tbW9Sx/K/IhsZNmzZBdru77jIqh+iu+Ajo9fxlGCtLLb3aC5Z86ymTtXldr3IpvtpvPEfyf5IrJa/xVOTOJaeDcsfWsSGz8TdWNNo/vwjG2CkyO1Z4GEsBXRWKgNYZyK2St1C5i4+X00GuG7tZT7DetI+EJge0bd261Z1DWes4TPnUmwaQU07loDvXTgOYIuANx3IooRxcqDct6Pf7IWdZFd0VLxP4VIwE9Hr+0rOVFUDyjZdxAsJLTyFIZbJxhk5zmapMc6fkdRYBjgSKTqwvZIg9/wUWubijMjkXVPMXJkd386ydJZOYGjsIsOdvB0XjYWjzx4k7k6lMNs4wXy7lf0OUvNTzLBCnYniBhevv2PgXvgjZyBS+DBJTQCGdSCT+mdNB+Rt95iwmTvvkDG1WAXO6ISt8WXvmdJ1BhPEygU9uIsCevztKS3qiespkryge81lK8j+hx5qK+9SlwJGAQWHpJGfs+TupNMylxZNLEM0hsuSaoy5L2O55Si0f+MVpBNjzd1qJWE+PNk/thc1I1iml90n9S3o+Rr9SMWyUVAHdsfEvIPw8RK01ZokH2XFncjJ8TXhyJVYyG6tvOB2UxSgq11457ZNrws4LX29aw/UHlNmAmUpeGyCmCsKq9KC/3BFgzz93bN0SMpXJKrKbnUre3NdYjgRSSccCvGfPvwDQXRCl15TJZ86cwalTp+KOa5ZzmuRY75qaGp5xZXOdpRCwGaiV4Nj4W6HmPT+3b9/G2NgY3nrrLQQCgSgAaSD37NmDyspK1zaQIuiGh4fR2dkZtecgGZTDDaurq7F69epofnljLwEKAXt5mgqNjb8pXHQcQ0Dqzttvv43+/v6IUNA+tbe3Y+PGja5oNDWh1t3djSNHjmhZQFNTExoaGrBu3TpPGDSKZrxANxQCBQDPxr8A0Is4SlEmj46OJlmGE2tX0pg6zTKcpuRtbm6Olooov2nFLYojvze5VzswBo0AFb4aCf7mgoCmTNaWT8ohafLOyHUnFFCliLVfq3dfoVqHrhkJLsmN1H2xWJe4DFbbyZvkgS/yRoD7BPKAmo1/HiAzijgCExMTKhwOx73L+DA9pFp9UGgMqsk4x7Pq5sVu1Vi6Xw1Nz8Z9yfQgpjplv0OsZTYRBCIQ5P+CV+EJ3JffcYe3YuO0j7fK20m5LSkpMZ2cuV9/hAtTPvjX/xd8Ic73YjxQ9oeo/dYXUblscdyXVA9imYtK3lR0nPWeQiAH5cHGPwdQGWSOCczh1pUP8AtU45XqL+FeUz+HG8NvoOt3duO5ilXY/twqw2l4//33I6t8qOQ1jKxgDqkYthE9G38bYTKoPBP4GMP7nsKGC3sQPrETZYvncOvyAL7bfAFP9r2MWoMjAC3RsvLn+vXrsDIi0cLgb34IcCRgA2c2/jZAZBCFJTD3MT66MAkM7sLKz+1aSIu/C7seMDYFtOAJkRVJFACxRJx7TyGQRdmw8c8CHr06isDcB/+AvxsE/F0XcWLnKizGZ5ga/h52jz2KL5uXAY7KGxOTngCFQHo+ul/Z+Oti4UvXEriL37x/FoNYicCjJfP6gM/jCyWrsAFfBBsJ1xasoYSzfA1huueIjb8JWHTqIgL/hovvhQCfH+WPLI2me3HZn+C5suijqRvZEPbRRx+5+igLUxl2sWMO9AwUnii5ZLnbgw8+GLcrU841EeWX7Mxcvny5gZDohAQcSODGP+HksfPAE5X4qkkFcKrctLa24sknn4z8z7z66quQXc28nEmAq4PSlIs0/idOnMBrr72WdKjV1q1b2fCnYcdPbiHwGa72P4+qLe/gieDP0FP/sC0Jp10EWzDmJ5DC71dzXgpkq73saJRt97E7HTs6OrjL0XnFxRRZJXAnpAKlCcdDJO0Wthr4PX+JR1lo/09ytIXsJjZ6rEV2qaDvdAQ4EoiRtez5x8DgLQnYTEDPLoJEIdOqfr8fZWUWFRA2p9NrwVEIAGDj77Vqz/wWmoAYjtGzi0DDMfkvGU8LATb++a9wjJEEYgnIiruRkRHIWUMiFLRL7CKsX78ea9eu1V7xN0cEPCkE2PjnqDYxWBLIgkA6uwi1tbU8giILtum8ekoIpGr8W1paItaMOCeZrqrwGwnkh4D8n54+fRp9fX1JFsd27NiB8vJyxxnKyQ+Z3MTiCSHAxj83lYehkkCuCVCZnGvCQFELATb+ua9AjIEE8kWAyuTckC5KIcDGPzeVhaGSgBMIUJlsbykUlRBg429v5WBoJOB0AlQmZ19CRSEE2PhnXxEYAgm4mYC0AWNjY+ju7k5SJjc0NGDdunVUJqcoYFcLATb+KUqVr0nAwwREmTw8PEwbxwbrgCuFABt/g6VLZyTgcQKiTD516hTa2tqiJOrq6sCdyVEc7lodxMZ/oeB4RwIkYJxAKmWy7BESoeDlncmuGAmw8Tde2emSBEggPQExeDM4OBhnG6SqqioyOti8ebPndiY7Wgiw8U9fmfmVBEjAOgFpX6hMduhmMTb+1is2fZIACZgn4GVlsqNGAmz8zVde+iABErCXgNeUyY4QAmz87a3EDI0ESCB7Al5RJhdUCLDxz76iMgQSIIHcE3CUMvnueRxcVYmWD9Pn29c6hMkDtekdydd0tidz9S2VDd+WlhYVDodzFS3DJQESIIGsCEjbJbaRxUayZi9Zfuvq6tTJkyfzaDP5mhpqrVDAN1Vw8k58nm7+QnU1+lXr0LX49yme8ioE2PinKAW+JgEScB2B69evq56eHlVVVRUnENrb29X4+Hhu8zN7UXX5fQr+LhWeTYzqtgp3v6mGppM+JDqMPOdFCLDx12XPlyRAAkVCQBp9afwTRwciJERY2H5ND6lWn0/5uy6qaFM/PaT+/FBIJYwLMkadUyHAxj8jfzogARIoIgLS4Mu0kEwPxQoEmeqWaSRpE7O/ZtX00H7lQ4PqCt++F9zNiyrYutXwFFBsGnKiGKbCN7Muhi5IgASKm4CmTO7t7cW5c+cimbVnZ/InuHy0ESt39SUAbEBXuBc7y+5PeJ/hMVYiZHvPnn+2BOmfBEig2AhIu2irMllHHzA7Oaj2//EhFTI7F6SUsmU6iI1/sVVb5ocESCAXBGxRJk8GVSOgSgMx8/+zF1X3oZ+paQuJzmo6iNM+GYZZ/EwCJEACKQiIVbTjx48nHXNdX1+PTZs2Yfny5To+53Bj+CWs2jCIbUPv4EDtQzpuzL2yJATY+JuDTNckQAIkkIqA7EwOhUI4fPgw3nrrragz7Zjr8vLyGKtoH2N431PY8HolgpOdqPfdF3Vv9caUEGDjbxUz/ZEACZBAZgKplMlvvvkmVq9ejbmr/dhdtQVHnwhisqcevsxBZnRhWAj09/fjtddei2q5JWSRVE1NTSgrK8sYER2QAAmQAAkYIyAdbjnmWkYGIhj+5m+O4P99vwGVLSMxAXzTltFARiEgJ+oFAoGkYQob/5iy4C0JkAAJuJRAygklNv4uLVEmmwRIgARMEFis51aGH4m9f9nkILY4OfWjR4zvSIAESMCdBHSFgDT0AwMDGB0djTT8kjXZ8VZdXQ2xwSmjBF4kQAIkQALuJ5BRJyBZ1JsaklGBKIbXrl3rfgrMAQmQAAl4lIAhIaCxoTDQSPCXBEiABIqDgCkhoGWZwkAjwV8SIAEScDcBXZ1ApizJFFCizkDWs1JnkIkcv5MACZCAMQKyk1hOIH311VcTPEygf/tXsGjRouS/FdtxsP88puYSvKR7tHDeUJIXOSEv8fxseZb3vEiABEiABIwR0E4cFfsDsfYIkttSzbxkjQqEbt4LfHZSvXeoUfngUzWB99T824wR23KKqBYLhYFGgr8kQAIkYJzAxMSE6ujoiDNVKWYr5Z2+3fV5IeDbH29G8k5IBUqhkPg+TVJsFQJaPBQGGgn+kgAJkIA+Aen161khEyP20obK95TXfGPvax2KPz46YnbSAUJASziFgUaCvyRAAiRwj4CePWLp9ZuxRzwb7lJ+JNgYVp+qK8FvKR+gkoRDGvg5GQkkxkdhkEiEzyRAAl4ioBmTSdSdinF6EQrmrtsq3NWggIoFm8KzkyoUDKhGHxRqXlZDk58aDjIvQkBLDYWBRoK/JEACxU5AU/LK9E6sklcEgUwDiWCwdmn6gK2q9Xn/Qtg1raprIKQmZ82FammfQLrVRka+cZ+BEUp0QwIk4EYCqWwCNDY2Ro7dKSkpyS5bN4axb9UGHNs2hEsHarEsu9CQ15FAonziyCCRCJ9JgATcSEB69XpKXlnqKe1cWiWvyQzfCQVUaZI+wGQgMc4LMhJIFFx6IwMarEmkxGcSIAGnEbBmJzibXHyCy0cbsXIX0BXuxc6y+7MJ7J7fGIFQ8Fu9kYFIUv11sgVPLhNAAiTgQQKakldW9MTO9VtT8poDOHslqHaK8rc0oEJ3zPlN5bqg00GpEkVhkIoM35MACRSCQCYlr53TPfr5u6lCgZo4oQM0qK7wbX3nJt46Yjoo1XiG00SpyPA9CZBAPgjkXMmbj0xkiMPRQkBLO4WBRoK/JEACuSYgB7eFQiEcPnw4yba62FEpLy/HkiVLcp2MvIXvCiGg0aAw0EjwlwRIwG4C+Vfy2p0Da+G5SghoWezv78drr70WMXmpveNqIo0Ef0mABIwSuHLlCoaHh9HZ2RnXnrS3t2Pjxo1YvXq10aBc686VQkBo3759GydOnKAwcG3VY8JJoDAEpO0YGxtDd3c3jhw5Ek1EU1MTGhoasG7duqKa7olmMMWNa4WAlh8KA40Ef0mABNIR0JS8zc3NUWdVVVWwbSdvNFR33bheCGi4KQw0EvwlARLQCIiSd2RkJGKhS6wfapdMH4uSV6wkev0qGiGgFSSFgUaCvyTgXQKyiOTUqVNoa2uLQsX4gGYAABcgSURBVJBGX3r9NTU1WL58efS912+KTghoBUphoJHgLwl4g0AqJW9HR0fE/rkXlLxWSrpohYAGg8JAI8FfEig+AvL/TSVvduVa9EJAw0NhoJHgLwm4nwCVvPaVoWeEgIaMwkAjwV8ScBcBKnlzU16eEwIaRgoDjQR/ScDZBKjkzW35eFYIaFgpDDQS/CUB5xAQJe/AwEBkaee5c+eiCaOSN4rCthvPCwGNJIWBRoK/JFAYAvI/ePr0afT19SXt5N2xY0fRHdxWGMrJsVIIJDChMEgAwkcSyDEBObhtdHQUiTt59+7di9raWmRtkzfH6Xd78BQCKUqQwiAFGL4mARsIpFLyysFt69ev505eGxgbDYJCIAMpCoMMgPiZBEwQECWvHN8QCASivriTN4qiIDcUAgaxUxgYBEVnJJBAIJ2S1+/3o6ysLMEHH/NJgELAJG0KA5PA6NyTBOT/hEpelxS9CXvEdBpDQAxLB4NBVVVVFWf8uaWlRYXD4RiXvCUBNxG4oyaD34yr0wDmn7+mGgNBFZr8NGOGYv8v5L6np0dNTExk9EcH+SeA/EdZXDFSGBRXeTI3QmBWTQ/tVz5AlQZC6k4Eyqdq8r1O1eiDQs0hFbo5mxbV6Oioam9vV/LLy9kEOB1k04iN00Q2gWQwDiAwhxvDL2HVhkFsG3oHB2ofmk/TLZw/+AwqW26iNe69A5LMJFgmsNiyT3qMI7BkyRLU19dHzjAPBoMQi0VyySqIlStXorW1FXLoFS8ScD6BGfxy7P9iyufHk5Wx5+5/gulrN52ffKbQFAEKAVO4MjumMMjMiC4cTmDuCn7+0zDw+19ByQMLTcTc1f+DHx47DyQJB4fnh8lLS2ChhNM640ezBCgMzBKje6cQmPvgH/B3g1Pwrf4Sfi/SQnyGqfP9OPTiX+LolB/7f/At1Cxj0+GU8so2HSzJbAlm8E9hkAEQP9tKQHbiDg4OYvPmzdi9e7eFsOdw68oH+AW+hifQj7pFi7Bo0X/Aisr/hWvrv4fQ5I/xau1/RqaGo7e3F2vWrIkcACf7BHg5mICz9dbFlzquJiq+MnVCjrTVOAvLOaHq6uosLMu8poZaKxR8+9XQdPoVQOny3dTUFLfMVJ4ljVL/eTmLAJeIFqg8KAwKBL6IopV197L+PnZNvgiBjo4ONT4+bi2nd0IqUAoFf5cKW5cBkbglfZIWW9NnLVf0lYYAhUAaOPn4lEoYyBrr69ev5yMJjMNFBKS+SI9ar6d98uTJrHvas+Eu5YdP+bsuqixlQBzVVCMV2XDJeh6HKu8PFAJ5R64fYSphID0p/pPoM/PSW9mFLnUhdrpHetjyzraduLMfqeDOrymgRgVCN3OCV+qyNPwyVRWbF9lpL4KCV/4JUAjkn3naGCkM0uLx1Md8Nph3QgFVGj0eQo6JsH80kFh4eRFsiZHyOYkAhUASEme8oDBwRjkUIhVemzrJ9RRXIcrQTXHy2AgHr9ySpKU6jkJsrW7duhXLl8fu6HR4Zpi8lARkGeXw8DA6OzvhZZu65JCyiuTug5sklpfTypFB8ZU+e8Dpy9RrI6L0NHL3lSOB3MnXnIQsIwMxxL19+/a48DkyiMPh6Ac5Q0o2dCXa1G1sbIxs8qJN3fjiS2WKsqWlBWKVbO3atfEe+GSKAIWAKVzOcSz/GD/60Y/iGhJJHYWBc8ooNiVsyGJpWL+nALXOLpVPCoFUZFzynsLA2QUlNnVPnTqFtra2aEJpUzeKwvKNjIjHxsbQ3d2NI0eORMNpampCQ0MD1q1bBzmyhVdmAhQCmRm5wgWFgXOKKZ1N3erqaqxevdo5iS2ClKRSJre3t2Pjxo3knaGMKQQyAHLbZwqDwpSY9ExpU7cw7GNjvXDhAo4fP5408hJbH5s2beJqulhY8/cUAjpQiuEVhUF+SlEandHR0TjdjBgU2rt3L2pra0Elb37KITEWqf+hUAiHDx/GW2+9Ff2sKZPLy8s5XTRPhUIgWj2K84bCwP5yFaYjIyORY5JjGxiZfli/fj1Xq9iPPKsQNWWyHG+t7cEQQc3VWPNYc7f6lCE7iYAcQZB49ox24iTPJjJWUrJuXc64iT3zRs7A4SFoxvgV2lWqfRlShnYcvlfo/FmNnyOBrPoY7vPMkYG5Mkun5PX7/SgrKzMXIF07goD8H7z99ttJO7S9qEymEHBElcx/IigMUjOnkjc1m2L84nVlMoVAMdZqE3miMFiARSXvAgsv3sn/gheVyRQCXqztOnlOJQxk5Usxb8uXfFPJq1MhPP7KS8pkCgGPV/bE7CcKg3A4XHTz3tpuU1nZEwgEogi4kzeKgjfzBLS6krgzWerKnj17imJnMoUAq7suAW1oLMrPYrn0lLzaUkEqeYullHOXD/mfKEplstVlRfRHAm4gIMsCZflfojlDsdErSz7lOy8SMEtgfHxciR3w2OXCYu6zp6cnp+Zgky3AiRW4xL8K1Tp0zXCWaFnMMCo6dBOBQv2TuokR05o9gYJ0MqaHVKsPCo1BNRmXhVl182K3aizdr4amZ+O+pHvgdFDuRo8MOc8EtOF6f39/3FEBXlz7nWf0jA5AvqYb5y4fxdMrXwK6hnFi5yosjqU/dwk9b/waX3+uBsti36e7Tych+I0EzBDIxVA1U/zaLlC9nbwyDcTd0JkI8rvdBHJbJ2fV9NB+5UOD6grfnk+6vDukDoVuWsoKp4MsYaOnlARsHqqmimdiYiJyDIbMw2pzonIvR2OEw+FU3vieBPJKQDohoidI1EmJPkGmLM1f19RQa4WCv0uFIzM+s+pmOKha/X9uagooNl4KgVgavM+awGy4S/nhU/6uiyppVnL2ouo+9DM1bTGWgsy/WkwrvZFAIgFb9FSzF1WX3xft+GgdoAWhkBhr5mcKgcyM6MIwAfuHqhK1Lf88hvNAhySQWwLZdGaSO1mfqsmhl9UfB0LqjsVkUwhYBEdvegTsG6raP4zWSy/fkUBhCZib1ryjJoPfVECNCsTM/8+G/1YdMrEkNDHHFAKJRPhsnUCWQ1VNoSZr+KPDXCAyn0olr/VioU/nE9DqfvoFDvOdLJ+5JaCZcs8loumWTvGbKQLJS9c+w9Tw97B7bBMGvlOB+zKEJlv0xSiLGP7QdvJu3ryZ1rkycOPn4iKgt9Q5cnzL713BvlUb8PoTQUz21MNnU7Yz/V/aFA2DKX4Cd/Gb989iECsReLRkfu3y5/GFklXYgC9mFADCZ8mSJXjzzTfx7//+76D5v+KvMcyhPoHly5dHrJ6J5TM52fZf//VfUVb2JVzt/yscmypFY12VbQJAUsCRgH458K1pAh9jeN9T2HDMj6FLr6B2WdwWFtOh0QMJkMA8gbvncXBVJVo+jCHSaN9ogEIghitvsyBwYzgnQ9UsUkSvJEACBgiwu2YAEp1kIvAZrr7bn5OhaqaY+Z0ESCA7AhwJZMePvm0eqsr5K0uXLoXMi/IiARLIPQEKgdwzZgwGCcjqoGeffTZy+FtTUxN27NhBBbFBdnRGAlYJcDrIKjn6ywmBxx9/PLI89MiRI6iuro4sGe3s7ISY++NFAiRgPwGOBOxnyhCzJKCZ9NMz/ygm/SorKzldlCVjencpgal+bF+xBb06yfc1BtD57W9gc4Uv/nhpHbexrygEYmnw3nEE9DbOSCJpI8BxRcUE5YuAthJvaQChS99BxX3A3NQZvPHCn+L53i8iEPoBvlPxHw2nhtNBhlHRYSEIaBtnBgYGMD4+Hmn8JR1tbW147LHHsGbNGvT29kKEBS8SKCYCUqcHBwch06F6l29LOR6Z3+672LcWzd/ejlL8BIf+fgw39DykeEchkAIMXzuPwOrVq/Hiiy9iZmYGJ0+eRF1dXeSIie3bt+PBBx/E7t27cebMGch0Ei8ScCsBqcOvvvpqpE4/+eSTaG5ujtRrLT93fzmG4FQFtj35BzHWw+YwM/1bzGiOzPxmOlyI30nAyQT0TmGUw+doXMbJpca0JRKQeizGZ2KNJGn1ON74zG0V7mpQiLMsppSa/UgFd35NAeaMzEs6eIpoYmnw2bUERkdHld4pjMFgkGYmXVuqxZtw7eTQxFNz5VlOzZXvSZd2Um/MSaKzkyEVDDQqH3yqZv+gmkyy5pQUStwLKobNDJvo1hUEZC51ZGQkoiuQFUbaJcpkOaV07dq12iv+kkDeCchyZ5nrl2ke7TJ8au68UvjYE8/j2X/5GxwamQLgQ03r/8Sf/Y9NeMbkyiCJn0JAKwX+FiUBOYVxdHQ06R9u7969qK2t5THVRVnqzstUqo5JS0tLRLdlrGMyhxvDL2HVhkFsG3oHB2ofsiWjFAK2YGQgTicgyuLTp0+jr68PshFNu7gzWSPB31wQECXvqVOnIqvZtPBlQYMcE11TU2Nyv8stnD/4DCpb/hO6wr3YWXa/FmRWvxQCWeGjZzcSkPOJZMmpLC0VAzba1dHRAb/fj7KyMu0Vf0nANAGpX8PDw5GlnYn1S3bByyo3S9fcJRx9uha78ArCJ3aizKa1nRQClkqDnoqFgPTU9HYmW+upFQsV5sMsAW2Xe3d3d9JIs6GhAevWrYsYTTIb7oL7z3C1/3lUbfk+lgZCuGTAUt+C3/R3FALp+fCrRwikmrOlMtkjFcBiNrNS8hqNU++kXn+XbaMBCgGjBUF3niFAZbJnitpSRlN1GMwpeS1FnRNPFAI5wcpAi4FAOmWyPUP8YqDknTwU69QhhYB36jBzmgWBnCn7skgTveaegBcWEVAI5L4eMYYiI2Dvsr8ig1ME2Uk3AixGQ0cUAkVQaZmFwhAotrnhwlB0Tqxe1QVRCDinDjIlLiaQl1UiLubj1KSnEuReWhVGIeDU2sl0uZJA7teLuxKL4xJdrEpeK6ApBKxQox8SMECAymQDkPLoxAtKXis4KQSsUKMfEjBJgMpkk8Bscu41Ja8VbBQCVqjRDwlYJJBqDtqtG40sYsi5N68qea2ApRCwQo1+SMAGAlQm2wAxJggRsG+//Tb6+/sj50Fpn7yk5NXybOaXQsAMLbolgRwQSKVMliOH9+zZY8PhYzlItEOC1NjpHQIo7CorK00e1+yQjOUxGRQCeYTNqEggEwGtN9vZ2Rl3zLX0Zjdu3Gj9GOJMEbvsu56SV7POxePAzRUmhYA5XnRNAnkjIPPax48fTzJIUl9fj02bNnmuh6speQ8fPhw33UPDQNlVSQqB7PjRNwnknICMDkKhEBIbP02ZXF5enuVZ9TnPQlYR6AlD6fWLiVAvCsOsYOp4phDQgcJXJOBUApoyOdYqmjYNsnnz5qKxmaxNi+kpeTktZnPtVLxIgARcR2BmZkaNjo6qpqYmBSD6V1dXp06ePKnku/XrpgoFaqJhxoYfd+/br4amZ61Hk+BTy1NLS0tc3Fqerl+/nuCDj3YQgB2BMAwSIIHCEZDGsaenR1VVVcU1nu3t7cp6wzmrpof2Kx9KVWPwXxIyN60udu1Upa1Dajrhi5XHiYkJ1dHREZd+yYu8C4fDVoKkHxMEKARMwKJTEnA6gfHxcSWNv/TYpQdtfURwW4W7GhTQoLrCt5OyPRv+W3Vo6FrSe6MvJF0yYpE0xo4uZGQjIxzr6TaaArrTCFAnYPP0GoMjAScQkDl1uZYvX24xOR9jeN9T2HBhT4wt248x/Bf/G7/zF3+KivusBUslrzVuufRlsShzmSSGTQIkkC0B643/fMw3/gknj03C/8p/w1cWy7sbuNx/CAc+fRx9JlsNTcnLvQ/Zlmpu/JssztwkgqGSAAk4i8Dcrz/ChakpDO76Kj63S0ubD/6uRjygPab51Xbydnd348iRI1GX3AUdReGYGwoBxxQFE0ICTiHwCT4YfQeDaEBXuBc7y+4H5q5i+KW/wNijJYgMDDIk9dlnn41u6CrGJawZsu+qzxQCriouJpYE8kHgGt4/NQaU/ike/fL99yJc/CBKVv5X4JGlhhIg5/aUlZVBev7FvpnNEBAHO6Ji2MGFw6SRQEEI3BjGvlUbcGzbEC4dqMWygiSCkeaLgJGRXb7SwnhIgAQKTmAON0Lv4thUKZ74w0coAApeHrlPAIVA7hkzBhJwD4G5Cbz7wx9jCo+j7o987kk3U2qZAKeDLKOjRxIoLgJ3zx/EqsoWfBjNVikagz9DT/3D0Te8KT4CFALFV6bMEQkUnIAc/CaH3Hn12OuCF4CJBFAImIBFpyRAAsYIyImmYu1Lu7xy7LWWXzf9Ugi4qbSYVhJwEQGvHHvtoiLRTSqFgC4WviQBErCLAHcP20UyN+FQCOSGK0MlARLQIcBzhHSgFPgVhUCBC4DRk4BXCeidKCo7jKlMzm+NoBDIL2/GRgIkkEBARgdetqGcgCPvjxQCeUfOCEmABFIRoDI5FZncvacQyB1bhkwCJGCRgKZMlmWmgUAgGop2FHVlZWUWBnOiwfEGAIUAqwEJkICjCWjKZNmAFrv3oL29HRs3bsTq1asdnX6nJ45CwOklxPSRAAlECegpk8Vewd69e7Fp0yaODqKkjN9QCBhnRZckQAIOISDTRadPn8bhw4fjRgdNTU3YsWMHbRiYKCcKAROw6JQESMB5BK5cuYKBgYHIWUXnzp2LJFCzZub3+yPGbZyXauekiELAOWXBlJAACWRBgMpka/AoBKxxoy8SIAEHE6Ay2XjhUAgYZ0WXJEACLiRAZXL6QqMQSM+HX0mABIqEAJXJ+gVJIaDPhW9JgASKmACVyQuFSyGwwIJ3JEACHiOQTpnc2NiImpqaot97QCHgsUrP7JIACegTEGXyyMhIZKlp4s7k9evXY+3atfoeXf6WQsDlBcjkkwAJ2E9AlMmjo6Nobm6OBq7tTK6trUVJSUn0vdtvKATcXoJMPwmQQM4IaMrkvr4+HDlyJBpPMe1MphCIFitvSIAESCA1AT1lsrju6OiAm3cmUwikLnN+IQESIAFdAmfOnImcWZR4zLUblckUArpFzJckQAIkkJlAMSiTKQQylzNdkAAJkEBGAm5VJlMIZCxaOiABEiAB4wTcpkymEDBetnRJAiRAAqYIpFMmV1dXO8IqGoWAqSKlYxIgARKwRkCUyadOnUJbW1s0ALGZbEaZfPf8QayqbMGH0RD0birQOvQODtQ+pPcx6R2FQBISviABEiCB3BFIpUxuaWmBCIWMO5NvDGPfqg14/YkgJnvq4YsmdQ63Lh3DtzaFsX3sFdQuWxz9ku6GQiAdHX4jARIggRwSuHz5MgYHB5N2JsvoYPPmzbo7k+cuH8XTK18CuoZxYucqxDX1c5fQ88av8fXnarDMYLopBAyCojMSIAESyBUB7SC77u7upJ3JDQ0NWLduHZYsWQJgDjeGX8KqDR/glXAvdpbdP//uDXT9zm48V/GA6SRSCJhGRg8kQAIkkDsCokweHh5GZ2cnNJvJ4+Pj80rkjzG87ylsuLAH4RM7UbZ4DrcuD+C7zRfwZN/LhqeAYlNPIRBLg/ckQAIk4CACokz+zW9+g/r6+nupmruEo0/XYtfgVHwq/V3zQiH+tZGn+4w4ohsSIAESIIH8E0hUEs998A/4u0HA33VxXh/wGaaGv4fdY4/iy3HKAeNppRAwzoouSYAESKCABO7iN++fxSBWIvBoybxC+PP4QskqbMAXYbUxt+qvgCAYNQmQAAl4kcC/4eJ7IcDnR/kjS6MAFpf9CZ4riz6avrE4gDAdDz2QAAmQAAlkQ+DGP+HksfPAE5X4qsE9AEaioxAwQoluSIAESKCgBD7D1Xf7cWyqFI11VTEbxLJPFFcHZc+QIZAACZBA7gjcPY+DqyrREntWRGPibmHr0VMIWGdHnyRAAiTgegKcDnJ9ETIDJEACJGCdAIWAdXb0SQIkQAKuJ0Ah4PoiZAZIgARIwDoBCgHr7OiTBEiABFxPgELA9UXIDJAACZCAdQL/H30TTrlgtiJvAAAAAElFTkSuQmCC"></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the probability that Iqbal passes at least two of the papers he attempts.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that Iqbal passes his third paper, given that he passed only one previous paper.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>A random variable <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span> is normally distributed with mean <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu ">
<mi>μ<!-- μ --></mi>
</math></span> and standard deviation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sigma ">
<mi>σ<!-- σ --></mi>
</math></span>, such that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(X < 30.31) = 0.1180">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo><</mo>
<mn>30.31</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0.1180</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(X > 42.52) = 0.3060">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>></mo>
<mn>42.52</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0.3060</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu ">
<mi>μ</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sigma ">
<mi>σ</mi>
</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>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {\left| {X - \mu } \right| < 1.2\sigma } \right)">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mo>|</mo>
<mrow>
<mi>X</mi>
<mo>−</mo>
<mi>μ</mi>
</mrow>
<mo>|</mo>
</mrow>
<mo><</mo>
<mn>1.2</mn>
<mi>σ</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A café serves sandwiches and cakes. Each customer will choose one of the following three options; buy only a sandwich, buy only a cake or buy both a sandwich and a cake.</p>
<p>The probability that a customer buys a sandwich is 0.72 and the probability that a customer buys a cake is 0.45.</p>
</div>
<div class="specification">
<p>Find the probability that a customer chosen at random will buy</p>
</div>
<div class="specification">
<p>On a typical day 200 customers come to the café.</p>
</div>
<div class="specification">
<p>It is known that 46 % of the customers who come to the café are male, and that 80 % of these buy a sandwich.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>both a sandwich and a cake.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>only a sandwich.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the expected number of cakes sold on a typical day.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that more than 100 cakes will be sold on a typical day.</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>A customer is selected at random. Find the probability that the customer is male and buys a sandwich.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>A female customer is selected at random. Find the probability that she buys a sandwich.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>Packets of biscuits are produced by a machine. The weights <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span>, in grams, of packets of biscuits can be modelled by a normal distribution where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X \sim {\text{N}}(\mu ,{\text{ }}{\sigma ^2})">
<mi>X</mi>
<mo>∼<!-- ∼ --></mo>
<mrow>
<mtext>N</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>μ<!-- μ --></mi>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<msup>
<mi>σ<!-- σ --></mi>
<mn>2</mn>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</math></span>. A packet of biscuits is considered to be underweight if it weighs less than 250 grams.</p>
</div>
<div class="specification">
<p>The manufacturer makes the decision that the probability that a packet is underweight should be 0.002. To do this <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu ">
<mi>μ<!-- μ --></mi>
</math></span> is increased and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sigma ">
<mi>σ<!-- σ --></mi>
</math></span> remains unchanged.</p>
</div>
<div class="specification">
<p>The manufacturer is happy with the decision that the probability that a packet is underweight should be 0.002, but is unhappy with the way in which this was achieved. The machine is now adjusted to reduce <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sigma ">
<mi>σ<!-- σ --></mi>
</math></span> and return <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu ">
<mi>μ<!-- μ --></mi>
</math></span> to 253.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu = 253">
<mi>μ</mi>
<mo>=</mo>
<mn>253</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sigma = 1.5">
<mi>σ</mi>
<mo>=</mo>
<mn>1.5</mn>
</math></span> find the probability that a randomly chosen packet of biscuits is underweight.</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>Calculate the new value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu ">
<mi>μ</mi>
</math></span> giving your answer correct to two decimal places.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the new value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sigma ">
<mi>σ</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>A continuous random variable <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> has the probability density function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> given by</p>
<p style="padding-left: 210px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced open="{" close><mtable columnalign="left"><mtr><mtd><mfrac><mi>x</mi><msqrt><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>k</mi></mrow></mfenced><mn>3</mn></msup></msqrt></mfrac><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mn>0</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>4</mn></mtd></mtr><mtr><mtd><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mn>0</mn><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mtext>otherwise</mtext></mtd></mtr></mtable></mfenced></math></p>
<p>where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>∈</mo><msup><mi mathvariant="normal">ℝ</mi><mo>+</mo></msup></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>16</mn><mo>+</mo><mi>k</mi></msqrt><mo>-</mo><msqrt><mi>k</mi></msqrt><mo>=</mo><msqrt><mi>k</mi></msqrt><msqrt><mn>16</mn><mo>+</mo><mi>k</mi></msqrt></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The following table below shows the marks scored by seven students on two different mathematics tests.</p>
<p><img src="data:image/png;base64,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"></p>
<p>Let <em>L</em><sub>1</sub> be the regression line of <em>x</em> on <em>y</em>. The equation of the line <em>L</em><sub>1</sub> can be written in the form <em>x</em> = <em>ay</em> + <em>b</em>.</p>
</div>
<div class="specification">
<p>Let <em>L</em><sub>2</sub> be the regression line of <em>y</em> on <em>x</em>. The lines <em>L</em><sub>1</sub> and <em>L</em><sub>2</sub> pass through the same point with coordinates (<em>p</em> , <em>q</em>).</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">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <em>p</em> and the value of <em>q</em>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Jennifer was absent for the first test but scored 29 marks on the second test. Use an appropriate regression equation to estimate Jennifer’s mark on the first test.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>In Lucy’s music academy, eight students took their piano diploma examination and achieved scores out of 150. For her records, Lucy decided to record the average number of hours per week each student reported practising in the weeks prior to their examination. These results are summarized in the table below.</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 Pearson’s product-moment correlation coefficient, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math>, for these data.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The relationship between the variables can be modelled by the regression equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>=</mo><mi>a</mi><mi>h</mi><mo>+</mo><mi>b</mi></math>. Write down the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>One of these eight students was disappointed with her result and wished she had practised more. Based on the given data, determine how her score could have been expected to alter had she practised an extra five hours per week.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Lucy asserts that the number of hours a student practises has a direct effect on their final diploma result. Comment on the validity of Lucy’s assertion.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Lucy suspected that each student had not been practising as much as they reported. In order to compensate for this, Lucy deducted a fixed number of hours per week from each of the students’ recorded hours.</p>
<p>State how, if at all, the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> would be affected.</p>
<div class="marks">[1]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>In a city, the number of passengers, <em>X</em>, who ride in a taxi has the following probability distribution.</p>
<p><img src="data:image/png;base64,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"></p>
<p>After the opening of a new highway that charges a toll, a taxi company introduces a charge for passengers who use the highway. The charge is $ 2.40 per taxi plus $ 1.20 per passenger. Let<em> T</em> represent the amount, in dollars, that is charged by the taxi company per ride.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find E(<em>T</em>).</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that Var(<em>X</em>) = 0.8419, find Var(<em>T</em>).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>The mean number of squirrels in a certain area is known to be 3.2 squirrels per hectare of woodland. Within this area, there is a 56 hectare woodland nature reserve. It is known that there are currently at least 168 squirrels in this reserve.</p>
<p>Assuming the population of squirrels follow a Poisson distribution, calculate the probability that there are more than 190 squirrels in the reserve.</p>
</div>
<br><hr><br><div class="specification">
<p>The age, <em>L</em>, in years, of a wolf can be modelled by the normal distribution <em>L</em> ~ N(8, 5).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that a wolf selected at random is at least 5 years old.</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>Eight wolves are independently selected at random and their ages recorded.</p>
<p>Find the probability that more than six of these wolves are at least 5 years old.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>A biased coin is weighted such that the probability, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>, of obtaining a tail is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>6</mn></math>. The coin is tossed repeatedly and independently until a tail is obtained.</p>
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi></math> be the event “obtaining the first tail on an even numbered toss”.</p>
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mi>E</mi></mfenced></math>.</p>
</div>
<br><hr><br><div class="specification">
<p>The random variable<em> X</em> has a binomial distribution with parameters <em>n</em> and <em>p</em>.<br>It is given that E(<em>X</em>) = 3.5.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the least possible value of <em>n</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>It is further given that P(<em>X</em> ≤ 1) = 0.09478 correct to 4 significant figures.</p>
<p>Determine the value of <em>n</em> and the value of <em>p</em>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The continuous random variable <em>X</em> 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}} {3ax}&,&{0 \leqslant x < 0.5} \\ {a\left( {2 - x} \right)}&,&{0.5 \leqslant x < 2} \\ 0&,&{{\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>
<mn>3</mn>
<mi>a</mi>
<mi>x</mi>
</mrow>
</mtd>
<mtd>
<mo>,</mo>
</mtd>
<mtd>
<mrow>
<mn>0</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</mi>
<mo><</mo>
<mn>0.5</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mi>a</mi>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mo>−<!-- − --></mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mo>,</mo>
</mtd>
<mtd>
<mrow>
<mn>0.5</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</mi>
<mo><</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mo>,</mo>
</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> </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="a = \frac{2}{3}"> <mi>a</mi> <mo>=</mo> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </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>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {X < 1} \right)"> <mrow> <mtext>P</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>X</mi> <mo><</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {s < X < 0.8} \right) = 2 \times {\text{P}}\left( {2s < X < 0.8} \right)"> <mrow> <mtext>P</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>s</mi> <mo><</mo> <mi>X</mi> <mo><</mo> <mn>0.8</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>2</mn> <mo>×</mo> <mrow> <mtext>P</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>s</mi> <mo><</mo> <mi>X</mi> <mo><</mo> <mn>0.8</mn> </mrow> <mo>)</mo> </mrow> </math></span>, and that 0.25 < <em>s</em> < 0.4 , find the value of <em>s</em>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>A continuous 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"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = \left\{ {\begin{array}{*{20}{l}} {\frac{{{x^2}}}{a} + b,}&{0 \leqslant x \leqslant 4} \\ 0&{{\text{otherwise}}} \end{array}} \right.{\text{where }}a{\text{ and }}b{\text{ are positive constants.}}">
<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>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mi>a</mi>
</mfrac>
<mo>+</mo>
<mi>b</mi>
<mo>,</mo>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>0</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</mi>
<mo>⩽<!-- ⩽ --></mo>
<mn>4</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>
<mrow>
<mtext>where </mtext>
</mrow>
<mi>a</mi>
<mrow>
<mtext> and </mtext>
</mrow>
<mi>b</mi>
<mrow>
<mtext> are positive constants.</mtext>
</mrow>
</math></span></p>
<p>It is given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(X \geqslant 2) = 0.75">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>⩾<!-- ⩾ --></mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0.75</mn>
</math></span>.</p>
</div>
<div class="specification">
<p>Eight independent observations of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span> are now taken and the random variable <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="Y">
<mi>Y</mi>
</math></span> is the number of observations such that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X \geqslant 2">
<mi>X</mi>
<mo>⩾<!-- ⩾ --></mo>
<mn>2</mn>
</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="a = 32"> <mi>a</mi> <mo>=</mo> <mn>32</mn> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b = \frac{1}{{12}}"> <mi>b</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mn>12</mn> </mrow> </mfrac> </math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{E}}(X)"> <mrow> <mtext>E</mtext> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{Var}}(X)"> <mrow> <mtext>Var</mtext> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the 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">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{E}}(Y)"> <mrow> <mtext>E</mtext> </mrow> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(Y \geqslant 3)"> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>⩾</mo> <mn>3</mn> <mo stretchy="false">)</mo> </math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="question">
<p>Rachel and Sophia are competing in a javelin-throwing competition.</p>
<p>The distances, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math> metres, thrown by Rachel can be modelled by a normal distribution with mean <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>56</mn><mo>.</mo><mn>5</mn></math> and standard deviation <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math>.</p>
<p>The distances, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi></math> metres, thrown by Sophia can be modelled by a normal distribution with mean <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>57</mn><mo>.</mo><mn>5</mn></math> and standard deviation <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>8</mn></math>.</p>
<p>In the first round of competition, each competitor must have five throws. To qualify for the next round of competition, a competitor must record at least one throw of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>60</mn></math> metres or greater in the first round.</p>
<p>Find the probability that only one of Rachel or Sophia qualifies for the next round of competition.</p>
</div>
<br><hr><br><div class="specification">
<p>The random variable <em>X</em> has a normal distribution with mean <em>μ</em> = 50 and variance <em>σ </em><sup>2</sup> = 16 .</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the probability density function for<em> X</em>, and shade the region representing P(<em>μ</em> − 2σ < <em>X</em> < <em>μ</em> + σ).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of P(<em>μ</em> − 2σ < <em>X</em> < <em>μ</em> + σ).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <em>k</em> for which P(<em>μ</em> − <em>k</em>σ < <em>X</em> < <em>μ</em> + <em>k</em>σ) = 0.5.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Timmy owns a shop. His daily income from selling his goods can be modelled as a normal distribution, with a mean daily income of $820, and a standard deviation of $230. To make a profit, Timmy’s daily income needs to be greater than $1000.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the probability that, on a randomly selected day, Timmy makes a profit.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The shop is open for 24 days every month.</p>
<p>Calculate the probability that, in a randomly selected month, Timmy makes a profit on between 5 and 10 days (inclusive).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider two events <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A">
<mi>A</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="B">
<mi>B</mi>
</math></span> such that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A) = k,{\text{ P}}(B) = 3k,{\text{ P}}(A \cap B) = {k^2}">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>k</mi>
<mo>,</mo>
<mrow>
<mtext> P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>3</mn>
<mi>k</mi>
<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>
<msup>
<mi>k</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A \cup B) = 0.5">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo>∪<!-- ∪ --></mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0.5</mn>
</math></span>.</p>
</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="k">
<mi>k</mi>
</math></span>;</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A' \cap B)">
<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">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The weights, in grams, of individual packets of coffee can be modelled by a normal distribution, with mean <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>102</mn><mo> </mo><mtext>g</mtext></math> and standard deviation <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo> </mo><mtext>g</mtext></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that a randomly selected packet has a weight less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn><mo> </mo><mtext>g</mtext></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The probability that a randomly selected packet has a weight greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math> grams is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>444</mn></math>. Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>A packet is randomly selected. Given that the packet has a weight greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>105</mn><mo> </mo><mtext>g</mtext></math>, find the probability that it has a weight greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>110</mn><mo> </mo><mtext>g</mtext></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>From a random sample of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>500</mn></math> packets, determine the number of packets that would be expected to have a weight lying within <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>5</mn></math> standard deviations of the mean.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Packets are delivered to supermarkets in batches of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>80</mn></math>. Determine the probability that at least <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20</mn></math> packets from a randomly selected batch have a weight less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>95</mn><mo> </mo><mtext>g</mtext></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question">
<p>It is given that one in five cups of coffee contain more than 120 mg of caffeine.<br>It is also known that three in five cups contain more than 110 mg of caffeine.</p>
<p>Assume that the caffeine content of coffee is modelled by a normal distribution.<br>Find the mean and standard deviation of the caffeine content of coffee.</p>
</div>
<br><hr><br><div class="specification">
<p>A random variable <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span> has a probability distribution given in the following table.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-02-28_om_17.11.47.png" alt="N16/5/MATHL/HP2/ENG/TZ0/01"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{E}}({X^2})">
<mrow>
<mtext>E</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>X</mi>
<mn>2</mn>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{Var}}(X)">
<mrow>
<mtext>Var</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The number of taxis arriving at Cardiff Central railway station can be modelled by a Poisson distribution. During busy periods of the day, taxis arrive at a mean rate of 5.3 taxis every 10 minutes. Let T represent a random 10 minute busy period.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that exactly 4 taxis arrive during T.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the most likely number of taxis that would arrive during T.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that more than 5 taxis arrive during T, find the probability that exactly 7 taxis arrive during T.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>During quiet periods of the day, taxis arrive at a mean rate of 1.3 taxis every 10 minutes.</p>
<p>Find the probability that during a period of 15 minutes, of which the first 10 minutes is busy and the next 5 minutes is quiet, that exactly 2 taxis arrive.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>There are 75 players in a golf club who take part in a golf tournament. The scores obtained on the 18th hole are as shown in the following table.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-09_om_16.43.55.png" alt="M17/5/MATHL/HP2/ENG/TZ2/01"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>One of the players is chosen at random. Find the probability that this player’s score was 5 or more.</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>Calculate the mean score.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The marks achieved by eight students in a class test are given in the following list.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="specification">
<p>The teacher increases all the marks by 2. Write down the new value for</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the mean.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the standard deviation.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the mean.</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>the standard deviation.</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>A ninth student also takes the test.</p>
<p>Explain why the median is unchanged.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>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 such that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A \cup B) = 0.95,{\text{ P}}(A \cap B) = 0.6">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo>∪<!-- ∪ --></mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0.95</mn>
<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>
<mn>0.6</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A|B) = 0.75">
<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>
<mn>0.75</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\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">[2]</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="{\text{P}}(A)">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that events <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A’">
<msup>
<mi>A</mi>
<mo>′</mo>
</msup>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="B">
<mi>B</mi>
</math></span> are independent.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>It is known that 56 % of Infiglow batteries have a life of less than 16 hours, and 94 % have a life less than 17 hours. It can be assumed that battery life is modelled by the normal distribution <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{N}}\left( {\mu ,\,\,{\sigma ^2}} \right)">
<mrow>
<mtext>N</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>μ<!-- μ --></mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<msup>
<mi>σ<!-- σ --></mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</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="\mu ">
<mi>μ</mi>
</math></span> and the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sigma ">
<mi>σ</mi>
</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>Find the probability that a randomly selected Infiglow battery will have a life of at least 15 hours.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Steffi the stray cat often visits Will’s house in search of food. 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 “the number of times per day that Steffi visits Will’s house”.</p>
<p>The random variable <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span> can be modelled by a Poisson distribution with mean 2.1.</p>
</div>
<div class="specification">
<p>Let<em> Y</em> be the discrete random variable “the number of times per day that Steffi is fed at Will’s house”. Steffi is only fed on the first four occasions that she visits each day.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that on a randomly selected day, Steffi does not visit Will’s house.</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>Copy and complete the probability distribution table for <em>Y</em>.</p>
<p style="text-align: center;"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAaIAAABdCAYAAADnuGIXAAAPdElEQVR4Ae3dD1CUZR4H8O+unnkGyHB6x642crKBzklakF7FeQix5Fw2Dk5SFkFwNWSZyWQUaTWnohlDWmNj14GinRcWOzXTnKbJcTd4XgxbejQhDHiQwtLocYibGcn73ry7QC/7B5Y/77v/vswwvLzv8z6/5/m8z/v+eP/wrkYURRH8ogAFKEABCnhJQOuluAxLAQpQgAIUsAkwEXEgUIACFKCAVwUmD0TXaDQDk/xJAQpQgAIUUFRAfldoMBFJEeULFG1BkFUuJflgt6UBQAMaSIc+jgO7gTwN8NKcXIPTFKAABSigugATkerkDEgBClCAAnIBJiK5BqcpQAEKUEB1ASYi1ckZkAIUoAAF5AJMRHINTlOAAhSggOoCTESqkzMgBShAAQrIBZiI5BqcpgAFKEAB1QWYiFQnZ0AKUIACFJALMBHJNThNAQpQgAKqCzARqU7OgBSgAAUoIBdgIpJrcJoCFKAABVQXYCJSnZwBKUABClBALsBEJNfgNAUoQAEKqC7ARKQ6OQNSgAIUoIBcgIlIrhFo04IF5vICLNNooNFn4fVaC4RA6+NI/bE2ocp0CMVZaSioujRS6cBbbm3C8eIs6KUxoNFjWcFBmC29gdfPYXvUC0vtHmTpBwxMaLIG3Z4gE+pFu+kp6NPK0OQjDExEss0TWJPdMJesxbONd+NQn4g+88O4WLAWJebuwOrmcL0RzqLsgU0or9yOjQf+O1zJwFwmtOGjd6qA+95Ehyiir+N93Ne5Ewlr9sAcNAdiAdYvjuAY0rGvY8DgZSRtqUZPYG71EXsltH+Ml57aA8uIJdUrwESknrWqkYQmEwpL5mPz8ynQaQGtLgXPb56PkkKTz/wVpDiIdh5yPj6MfS9vgFHxYL4XQPjPBUxfnYPUmDBb47S6u7D+xQ0wVv8Zh2u7fK/BSrRIaEPd5duQuVgH6WCn1d2B3KwVwMFPUdfjI6cDSvTbXZ3WWpQU/gMzUhe4K+GV+UxEXmFXOug1NNccxbE4A2aHDGxiLcIS7kZm/VHUNF9TugGs3wcEtNF3IWnWlCEt0UZGYZFuyKzA/kX7SyQl3WRLQvaO9qKztQ2x+SuxOGxg3whsgh971w3z3kPA02uRFnnDj7N9YMpPtsR5mLIMto/Y1WjuRfHg5aVLqCpIgD7XhPYg/OPG7fgRWlFTUQPdoihEOm3hGlTUtAbfvSK3WEG4YNqvsSTWfpYUVL2X7heW/QHbmrNxKH8xQoKq8wKs5gN4A2uQFx/hcz13Okz5XAttDboJ6eXN6GsshRGf4/iZzv4DaRjmp61A7IVufOubDfdOq6wdaKwH4mL1QbazeYfbf6IK6Kmrxum8R2B0OFPynz6MpaUCeqoKoQ+NRUrudhw49Tecag6yO0TWOux9A3g6L8Enjwl+kojsg087dyFSoy1o6fq2PxFNwS9mR2Fu6gLo/aonY9mZuA4FxilgrcMfy2egyEcPRuPs3TCraxGWXIQO8Xt01B3Ac9iPVUmFMLUHy9OD3TC/U4W5RXmIH7xUPwyXFxb51+FbeyMionVoqW/DRQlLaMOHJV/hdw8u8sks74XtaQ8ZokdsHFDf2AGr1xrBwL4lIB2MjiDihWyfPRgp7zUFuvhMbH97K4yWf+GzxmA4K5Iuyb2HyjkPYqUPnwX7XSIKj5zWP17tj2XWGdf6NLDyO5eLCNooJGYkOi0QOltx2pKIjMQo2c1bp2KcEXACvWj/6D18uXwDcuYF4b0hh+2pNdyJDKNv3ax3aOIE/tqF2sN/wvZVUZhk+18y6X+pZiJlpxk4lovYSXqklZ31+j1j/0pE+CnCI8OBk+dw/vwJ7KzU4cmVc3hQdRq2U2FIvAdxQx5RFWC90Ix64z1INEx1WoMzAlWgF5aqMhwOXYHMwSTUg7PlB1EdjI8vS5tZuofacmuQPLAxA8mv1kEURdn3RZx4Lh4wlqKxrwOf5Mzz+jHUzxLRVEyfGQpcPYk9u84gad1yzPKzHqh1uNPGpKMovwFbdpyARQAEywns2NKA/KJ0xNBMrc3g5Tg9aDK9gjUpTyA/ZbbsL+LpmH/oB+h99H7BRKIJ7Sbk6tNQUF5r2w8gtKNqx150FuYF2QMbE6k68XX52SFpMkIjZgKIwG8fexzJuqH/IzHxPP5cYzji89/CqzPfRfwkDSat+RSxxW8hPz7cnzs1yrbbH++fFJuLYzBjZ8pMaHzotSaj7Mwoi0uvcSlE0qrtqHZaUwdjxp0w+Nne79QND2Zop8fgrtQO7MxeAv0kDfSP/gXdq3ZjX84C3lf2wE+tIhpROmcDbP+j0z+pVuwxxLmGprIXsT8qH1uTZ3n9dNLTDmg0GttpsaflA7EcDfxlH1N29HEccBxII8xxHPjR30TS0x/7UYpHUehHSUjZ3Zq1U4ACFPB/gcm+3QUp+ezGihVtyNx3Mxq/uh2bN/CU2re3GVtHAQpQYHQCPn9GJFy+iEZLAxo7b8cz64PttRyj25gsTQEKUMAfBfzsHpE/EjtfD/XPXoyv1Y7XhMdXm3+uTQPuC9LI5ThwNvD5MyL/POSw1RSgAAUo4KkAE5GnUixHAQpQgAKKCDARKcLKSilAAQpQwFMBJiJPpViOAhSgAAUUEWAiUoSVlVKAAhSggKcCTESeSrEcBShAAQooIsBEpAgrK6UABShAAU8FmIg8lWI5ClCAAhRQRICJSBFWVkoBClCAAp4KMBF5KsVyFKAABSigiMCQV/woEoGVUoACFKAABRwE5B87NOTt2/IFDuvw13EI8N1Szu+WGgen367KccBxIA1ejgO7gXxH5qU5uQanKUABClBAdQEmItXJGZACFKAABeQCTERyDU5TgAIUoIDqAkxEqpMzIAUoQAEKyAWYiOQanKYABShAAdUFmIhUJ2dAClCAAhSQCzARyTU4TQEKUIACqgswEalOzoAUoAAFKCAXYCKSa3CaAhSgAAVUF2AiUp2cASlAAQpQQC7ARCTX4DQFKEABCqguwESkOjkDUoACFKCAXICJSK7BaQpQgAIUUF2AiUh1cgakAAUoQAG5gNcSkWCpRXnhG6jqEeTt8ZHpS6gq3gFTU4+PtIfNoAAFKBC4AuNMRNfQVLba9vka0mdsDPnWZ6H4g2o0WZ0TjWA5jk3Pf4zQ7Gwkh0lNuA6LKW/o+oZimK9bYS5eJptvQJbpvApbYwaSHvoN2rb9HoVV7XDugQpNYAgKUIACwSIg9n8BGJgc/c++BrHUGC0aSxvEPtval8XGyhfEJEDUPbJfbLhin2tbdOUz8TXjerHywvcOcfrEK3UltnWiX6sTfxhc+p3YWHq/iKQXxMrGy4NzVZm4Ui+W5qwXSxvGF3dctqp0VPkgNBBFGtBA2tM4DpwNxnlG5C5dhyEm/SW8XXo/LAfexL7arv6C19B0eDeOZ+Rh5awpDitrEXLrcmQadWipb8NF21IB1rMV2Fb0c1S++wrSY8Ic1lH415AFyC64BRXrP0ATT4sUxmb1FKBAsAoolIgkzimIjDJAhw6cbr1kv7zV80+UbvoBGYlRcBlYOxsLU2OBk+fQcR2AtQ57n/gQcRVbke6UuNTZZFrDnchAKUqrL6kTkFEoQAEKBJmAy3wwMQa96GxthgV6LIqaAS0E9NR9ioNx9yDRMNVNiGm4+bY7oGtpRts3bagqKkZ95hbkxYe7KT/G2RYTsgbuaS17HeaB+1g9VSjQxyHX1PbjfSFtFBIzZuHgJ/8GH10YozdXowAFKDCMwAQmoquo//tn/Q8n9MJSW4ptm94Hkh7C6sURALpQ98kxYFEUIt1G1SJsfgJScRKVhc/g8W8eQFH2AoS47MB5mLIMsgcZHB6W6E80hmIzpJOrIV+6dJSL36Gx9H6g+hTOdPTaF4fEIi17Di50fzekuPSL5XQrOnl5zsmFMyhAAQqMV8BtShhLxZYD2ZgfOgkazQ3QL9kL5Fei7tCTiA/RAsIltJ7uGLnamXMQF/0lDn29FBW7V2KW2xbehPTyZukJi2G/m5+Nx2SXUadi7sLFiMZFdF3pT1Xan2G24VdIXRgpu3TYf4mxvhkXBs6cXNbHmRSgAAUoMBYB18fosdSEaTCWNuBIzjzZQXz0FQnnzuB4iw7GwuW4VUpgCn5pQyMQjfOob/sfEB8Cof2vKKldgs3ZLi4FWprR2tkLhLm7rKhgQ1k1BShAgQAWUPZIP2q4a2iuOYpjSHT/QMNgneO4NNdfhzY0HJGD9XXji8NNMG6+1/VZmM6AqEjHJ/0GV+YEBShAAQqMUWACz4hGaIF2BqIW6YcvJLSipqIGMG4d5oGGgSr6L82VD/w+hp83hiNS14LKc1/jfNUpVM55EFvdPZ0XZ8Bshc/QxtADrkIBClDA7wUm5ozoag+6eoHerh5cdUsSgYQ0I+D2pn8vLNUVOHjMgujUhZg7MS1z2xrbgmnTMXMacPWLMuyqXoh1K+e4uKxof/pv+Icshg/DpRSgAAUo4F5gnIf7/lf8hC7BxuoWVG9cglDNapQ1XXMRUYuwhLuRWX8UNc0Oy22PU98AfcorqAbQsjEBP7G94sdFNRM5S3sjIqJ1wOSleOy5FOhcadjO0tqRmXYLVP532onsKeuiAAUo4LMCmv5XTtgeg7a/fULJtkqJKxfrsHncDzVMSCuFsyh7rBJRWzciWef6/o/QVIbl64A3j+QgxlWi8qAh0jv4lLf1oCFeLEIDqLSPeXEjexCa44DjQBomjuNgjIdWD0acyyJTEbN6PVIrdmH/WW//e2g3zLsrgI3r3CYhCG348NUTSC1KH3MScsnAmRSgAAUoMCig8hmRPa7t7dvZlTDs3omceWpe8OqGufhhrLCkY9/sVny1NB8b3L21wXoWpi27UJe2GVuTZ7m4dzRoOOKEY/YfcYUALEAD578CA3Azj9gljgOOA2mQOI4Dlc+I7ONUq0tFUeUzCD+wX+XPI7qOyxc7Yfm8EZ1Ln8B6d0kIl1C1txJXVr887iQ04p7JAhSgAAWCXMArZ0TBZu6Y/YOt/1J/aUADjgP7ns99wXlf8MoZUTAeiNlnClCAAhRwLcBE5NqFcylAAQpQQCUBJiKVoBmGAhSgAAVcCzARuXbhXApQgAIUUEmAiUglaIahAAUoQAHXAkxErl04lwIUoAAFVBJgIlIJmmEoQAEKUMC1ABORaxfOpQAFKEABlQSYiFSCZhgKUIACFHAtwETk2oVzKUABClBAJYEhr/hRKSbDUIACFKBAkAvIPxpn8KPC5TOD3IfdpwAFKEABFQV4aU5FbIaiAAUoQAFngf8DryRfl5vMKHwAAAAASUVORK5CYII="></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the expected number of times per day that Steffi is fed at Will’s house.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>In any given year of 365 days, the probability that Steffi does not visit Will for at most <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span> days in total is 0.5 (to one decimal place). 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">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the expected number of occasions per year on which Steffi visits Will’s house and is not fed is at least 30.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question">
<p>Runners in an athletics club have season’s best times for the 100 m, which can be modelled by a normal distribution with mean 11.6 seconds and standard deviation 0.8 seconds. To qualify for a particular competition a runner must have a season’s best time of under 11 seconds. A runner from this club who has qualified for the competition is selected at random. Find the probability that he has a season’s best time of under 10.7 seconds.</p>
</div>
<br><hr><br><div class="specification">
<p>A discrete random variable <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span> follows a Poisson distribution <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{Po}}(\mu )">
<mrow>
<mtext>Po</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>μ<!-- μ --></mi>
<mo stretchy="false">)</mo>
</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}}(X = x + 1) = \frac{\mu }{{x + 1}} \times {\text{P}}(X = x),{\text{ }}x \in \mathbb{N}">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>=</mo>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mi>μ</mi>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mo>×</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>=</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">N</mi>
</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>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(X = 2) = 0.241667">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>=</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0.241667</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(X = 3) = 0.112777">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>=</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0.112777</mn>
</math></span>, use part (a) to find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu ">
<mi>μ</mi>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Each of the 25 students in a class are asked how many pets they own. Two students own three pets and no students own more than three pets. The mean and standard deviation of the number of pets owned by students in the class are <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{18}}{{25}}">
<mfrac>
<mrow>
<mn>18</mn>
</mrow>
<mrow>
<mn>25</mn>
</mrow>
</mfrac>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{24}}{{25}}">
<mfrac>
<mrow>
<mn>24</mn>
</mrow>
<mrow>
<mn>25</mn>
</mrow>
</mfrac>
</math></span> respectively.</p>
<p>Find the number of students in the class who do not own a pet.</p>
</div>
<br><hr><br>