File "markSceme-HL-paper2.html"
Path: /IB QUESTIONBANKS/5 Fifth Edition - PAPER/HTML/Math AA/Topic 4/markSceme-HL-paper2html
File size: 448.01 KB
MIME-type: text/html
Charset: utf-8
<!DOCTYPE html>
<html>
<meta http-equiv="content-type" content="text/html;charset=utf-8">
<head>
<meta charset="utf-8">
<title>IB Questionbank</title>
<link rel="stylesheet" media="all" href="css/application-02ef852527079acf252dc4c9b2922c93db8fde2b6bff7c3c7f657634ae024ff1.css">
<link rel="stylesheet" media="print" href="css/print-6da094505524acaa25ea39a4dd5d6130a436fc43336c0bb89199951b860e98e9.css">
<script src="js/application-9717ccaf4d6f9e8b66ebc0e8784b3061d3f70414d8c920e3eeab2c58fdb8b7c9.js"></script>
<script type="text/javascript" async src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/MathJax.js?config=TeX-MML-AM_CHTML-full"></script>
<!--[if lt IE 9]>
<script src='https://cdnjs.cloudflare.com/ajax/libs/html5shiv/3.7.3/html5shiv.min.js'></script>
<![endif]-->
<meta name="csrf-param" content="authenticity_token">
<meta name="csrf-token" content="iHF+M3VlRFlNEehLVICYgYgqiF8jIFlzjGNjIwqOK9cFH3ZNdavBJrv/YQpz8vcspoICfQcFHW8kSsHnJsBwfg==">
<link href="favicon.ico" rel="shortcut icon">
</head>
<body class="teacher questions-show">
<div class="navbar navbar-fixed-top">
<div class="navbar-inner">
<div class="container">
<div class="brand">
<div class="inner"><a href="http://ibo.org/">ibo.org</a></div>
</div>
<ul class="nav">
<li>
<a href="../../../../../../../index.html">Home</a>
</li>
<!-- - if current_user.is_language_services? && !current_user.is_publishing? -->
<!-- %li= link_to "Language services", tolk_path -->
</ul>
<ul class="nav pull-right">
<li class="dropdown">
<a class="dropdown-toggle" data-toggle="dropdown" href="#">
Help
<b class="caret"></b>
</a>
<ul class="dropdown-menu">
<li><a href="https://questionbank.ibo.org/video_tour?locale=en">Video tour</a></li>
<li><a href="https://questionbank.ibo.org/instructions?locale=en">Detailed instructions</a></li>
<li><a target="_blank" href="https://ibanswers.ibo.org/">IB Answers</a></li>
</ul>
</li>
<li>
<a href="https://06082010.xyz">IB Documents (2) Team</a>
</li></ul>
</div>
</div>
</div>
<div class="page-content container">
<div class="row">
<div class="span24">
<div class="pull-right screen_only"><a class="btn btn-small btn-info" href="https://questionbank.ibo.org/updates?locale=en">Updates to Questionbank</a></div>
<p class="muted language_chooser">
User interface language:
<a href="https://questionbank.ibo.org/en/users/set_user_locale?new_locale=en">English</a>
|
<a href="https://questionbank.ibo.org/en/users/set_user_locale?new_locale=es">Español</a>
</p>
</div>
</div>
<div class="page-header">
<div class="row">
<div class="span16">
<p class="back-to-list">
</p>
</div>
<div class="span8" style="margin: 0 0 -19px 0;">
<img style="width: 100%;" class="qb_logo" src="https://mirror.ibdocs.top/qb.png" alt="Ib qb 46 logo">
</div>
</div>
</div>
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>recognises that <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mn>0</mn><mi>m</mi></munderover><mtext>arccos</mtext><mo> </mo><mi>x</mi><mo> </mo><mo>d</mo><mi>x</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>5</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo> </mo><mtext>arccos</mtext><mo> </mo><mi>m</mi><mo>-</mo><msqrt><mn>1</mn><mo>-</mo><msup><mi>m</mi><mn>2</mn></msup></msqrt><mo>-</mo><mfenced><mrow><mn>0</mn><mo>-</mo><msqrt><mn>1</mn></msqrt></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>5</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>360034</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>360</mn></math> <em><strong>A1</strong></em></p>
<p><br><em><strong>[2 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>attempts to find at least one endpoint (limit) both in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> (or their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math>) and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>m</mi><mo>-</mo><mi>a</mi><mo>≤</mo><mi>X</mi><mo>≤</mo><mi>m</mi><mo>+</mo><mi>a</mi></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>3</mn></math> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mrow><mn>0</mn><mo>.</mo><mn>360034</mn><mo>…</mo><mo>-</mo><mi>a</mi></mrow><mrow><mn>0</mn><mo>.</mo><mn>360034</mn><mo>…</mo><mo>+</mo><mi>a</mi></mrow></munderover><mtext>arccos</mtext><mo> </mo><mi>x</mi><mo> </mo><mo>d</mo><mi>x</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>3</mn></math> <em><strong>(A1)</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>(A1)</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mrow><mi>m</mi><mo>-</mo><mi>a</mi></mrow><mrow><mi>m</mi><mo>+</mo><mi>a</mi></mrow></munderover><mtext>arccos</mtext><mo> </mo><mi>x</mi><mo> </mo><mo>d</mo><mi>x</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>3</mn></math>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mfenced open="[" close="]"><mrow><mi>x</mi><mo> </mo><mtext>arccos</mtext><mo> </mo><mi>x</mi><mo>-</mo><msqrt><mn>1</mn><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></msqrt></mrow></mfenced><mrow><mn>0</mn><mo>.</mo><mn>360034</mn><mo>…</mo><mo>-</mo><mi>a</mi></mrow><mrow><mn>0</mn><mo>.</mo><mn>360034</mn><mo>…</mo><mo>+</mo><mi>a</mi></mrow></msubsup></math></p>
<p>attempts to solve their equation for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> <em><strong>(M1)</strong></em></p>
<p><strong><br>Note:</strong> The above <em><strong>(M1)</strong></em> is dependent on the first <em><strong>(M1)</strong></em>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>124861</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>125</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mrow><mo>-</mo><mi>a</mi></mrow><mi>a</mi></munderover><mtext>arccos </mtext><menclose notation="left"><mi>x</mi><mo>-</mo><mn>0</mn><mo>.</mo><mn>360034</mn><mo>…</mo><menclose notation="left"><mo> </mo><mo>d</mo><mi>x</mi><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>0</mn><mo>.</mo><mn>3</mn></mrow></mfenced></menclose></menclose></math> <em><strong>(M1)(A1)</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Only award <em><strong>(M1)</strong></em> if at least one limit has been translated correctly.</p>
<p><strong>Note:</strong> Award <em><strong>(M1)(A1)</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mrow><mo>-</mo><mi>a</mi></mrow><mi>a</mi></munderover><mtext>arccos </mtext><menclose notation="left"><mi>x</mi><mo>-</mo><mi>m</mi><menclose notation="left"><mo> </mo><mo>d</mo><mi>x</mi><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>0</mn><mo>.</mo><mn>3</mn></mrow></mfenced></menclose></menclose></math>.</p>
<p><br>attempts to solve their equation for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>124861</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>125</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p><strong>EITHER </strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mrow><mo>-</mo><mi>a</mi></mrow><mi>a</mi></munderover><mtext>arccos </mtext><mfenced><mrow><mi>x</mi><mo>+</mo><mn>0</mn><mo>.</mo><mn>360034</mn><mo>…</mo></mrow></mfenced><mo> </mo><mo>d</mo><mi>x</mi><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>0</mn><mo>.</mo><mn>3</mn></mrow></mfenced></math> <em><strong>(M1)(A1)</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Only award <em><strong>(M1)</strong></em> if at least one limit has been translated correctly.</p>
<p><strong>Note:</strong> Award <em><strong>(M1)(A1)</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mrow><mo>-</mo><mi>a</mi></mrow><mi>a</mi></munderover><mtext>arccos </mtext><mfenced><mrow><mi>x</mi><mo>+</mo><mi>m</mi></mrow></mfenced><mo> </mo><mo>d</mo><mi>x</mi><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>0</mn><mo>.</mo><mn>3</mn></mrow></mfenced></math>.</p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mrow><mn>2</mn><mfenced><mrow><mn>0</mn><mo>.</mo><mn>360034</mn><mo>…</mo></mrow></mfenced><mo>-</mo><mi>a</mi></mrow><mrow><mn>2</mn><mfenced><mrow><mn>0</mn><mo>.</mo><mn>360034</mn><mo>…</mo></mrow></mfenced><mo>+</mo><mi>a</mi></mrow></munderover><mtext>arccos </mtext><mfenced><mrow><mi>x</mi><mo>-</mo><mn>0</mn><mo>.</mo><mn>360034</mn><mo>…</mo></mrow></mfenced><mo> </mo><mo>d</mo><mi>x</mi><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>0</mn><mo>.</mo><mn>3</mn></mrow></mfenced></math> <em><strong>(M1)(A1)</strong></em></p>
<p><strong><br>Note:</strong> Only award <em><strong>(M1)</strong></em> if at least one limit has been translated correctly.</p>
<p><strong>Note:</strong> Award <em><strong>(M1)(A1)</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mrow><mn>2</mn><mi>m</mi><mo>-</mo><mi>a</mi></mrow><mrow><mn>2</mn><mi>m</mi><mo>+</mo><mi>a</mi></mrow></munderover><mtext>arccos </mtext><mfenced><mrow><mi>x</mi><mo>-</mo><mi>m</mi></mrow></mfenced><mo> </mo><mo>d</mo><mi>x</mi><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>0</mn><mo>.</mo><mn>3</mn></mrow></mfenced></math>.</p>
<p><br><strong>THEN</strong></p>
<p>attempts to solve their equation for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> <em><strong>(M1)</strong></em></p>
<p><strong><br>Note:</strong> The above <em><strong>(M1)</strong></em> is dependent on the first <em><strong>(M1)</strong></em>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>124861</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>125</mn></math> <em><strong>A1</strong></em></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;">
[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>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>
<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="\left( {{\text{P}}\left( {1 < X < 3} \right) = } \right)\int_1^2 {3a\,{\text{d}}x + a} \int_2^3 { - {x^2} + 6x - 5} \,{\text{d}}x"> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>P</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo><</mo> <mi>X</mi> <mo><</mo> <mn>3</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> </mrow> <mo>)</mo> </mrow> <msubsup> <mo>∫</mo> <mn>1</mn> <mn>2</mn> </msubsup> <mrow> <mn>3</mn> <mi>a</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>+</mo> <mi>a</mi> </mrow> <msubsup> <mo>∫</mo> <mn>2</mn> <mn>3</mn> </msubsup> <mrow> <mo>−</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>6</mn> <mi>x</mi> <mo>−</mo> <mn>5</mn> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </math></span> <em><strong>(M1)(A1)(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 3a + \frac{{11}}{3}a"> <mo>=</mo> <mn>3</mn> <mi>a</mi> <mo>+</mo> <mfrac> <mrow> <mn>11</mn> </mrow> <mn>3</mn> </mfrac> <mi>a</mi> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{20}}{3}a\,\left( { = 6.67a} \right)"> <mo>=</mo> <mfrac> <mrow> <mn>20</mn> </mrow> <mn>3</mn> </mfrac> <mi>a</mi> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mn>6.67</mn> <mi>a</mi> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></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><img src="data:image/png;base64,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"> <em><strong>A4</strong></em></p>
<p>award <em><strong>A1</strong></em> for (0, 3<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span>), <em><strong>A1</strong></em> for continuity at (2, 3<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span>), <em><strong>A1</strong></em> for maximum at (3, 4<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span>), <em><strong>A1</strong></em> for (5, 0)</p>
<p><strong>Note:</strong> Award <em><strong>A3</strong></em> if correct four points are not joined by a straight line and a quadratic curve.</p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {0 \leqslant X \leqslant 5} \right) = 6a + a\int_2^5 { - {x^2} + 6x - 5} \,{\text{d}}x"> <mrow> <mtext>P</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mo>⩽</mo> <mi>X</mi> <mo>⩽</mo> <mn>5</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>6</mn> <mi>a</mi> <mo>+</mo> <mi>a</mi> <msubsup> <mo>∫</mo> <mn>2</mn> <mn>5</mn> </msubsup> <mrow> <mo>−</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>6</mn> <mi>x</mi> <mo>−</mo> <mn>5</mn> </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=" = 15a"> <mo>=</mo> <mn>15</mn> <mi>a</mi> </math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="15a = 1"> <mn>15</mn> <mi>a</mi> <mo>=</mo> <mn>1</mn> </math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow a = \frac{1}{{15}}\left( { = 0.0667} \right)"> <mo stretchy="false">⇒</mo> <mi>a</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mn>15</mn> </mrow> </mfrac> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mn>0.0667</mn> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</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) = \frac{1}{5}\int_0^2 {x\,{\text{d}}x} + \frac{1}{{15}}\int_2^5 { - {x^3} + 6{x^2} - 5} x\,{\text{d}}x"> <mrow> <mtext>E</mtext> </mrow> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> <msubsup> <mo>∫</mo> <mn>0</mn> <mn>2</mn> </msubsup> <mrow> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> <mo>+</mo> <mfrac> <mn>1</mn> <mrow> <mn>15</mn> </mrow> </mfrac> <msubsup> <mo>∫</mo> <mn>2</mn> <mn>5</mn> </msubsup> <mrow> <mo>−</mo> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> <mo>+</mo> <mn>6</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>5</mn> </mrow> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </math></span> <em><strong>(M1)(A1)</strong></em></p>
<p>= 2.35 <em><strong>A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to use <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^m {f\left( x \right)} {\text{d}}x = 0.5"> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>m</mi> </msubsup> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> <mn>0.5</mn> </math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0.4 + a\int_2^m { - {x^2} + 6x - 5} \,{\text{d}}x = 0.5"> <mn>0.4</mn> <mo>+</mo> <mi>a</mi> <msubsup> <mo>∫</mo> <mn>2</mn> <mi>m</mi> </msubsup> <mrow> <mo>−</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>6</mn> <mi>x</mi> <mo>−</mo> <mn>5</mn> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <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="a\int_2^m { - {x^2} + 6x - 5} \,{\text{d}}x = 0.1"> <mi>a</mi> <msubsup> <mo>∫</mo> <mn>2</mn> <mi>m</mi> </msubsup> <mrow> <mo>−</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>6</mn> <mi>x</mi> <mo>−</mo> <mn>5</mn> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> <mn>0.1</mn> </math></span></p>
<p>attempt to solve integral using GDC and/or analytically <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{{15}}\left[ { - \frac{1}{3}{x^3} + 3{x^2} - 5x} \right]_2^m = 0.1"> <mfrac> <mn>1</mn> <mrow> <mn>15</mn> </mrow> </mfrac> <msubsup> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> <mo>+</mo> <mn>3</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>5</mn> <mi>x</mi> </mrow> <mo>]</mo> </mrow> <mn>2</mn> <mi>m</mi> </msubsup> <mo>=</mo> <mn>0.1</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m = 2.44"> <mi>m</mi> <mo>=</mo> <mn>2.44</mn> </math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">c.iii.</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>
<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">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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>E</mtext><mfenced><mi>X</mi></mfenced><mo>=</mo><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><munderover><mo>∫</mo><mn>0</mn><mn>1</mn></munderover><msup><mi>x</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>d</mo><mi>x</mi></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><msubsup><mfenced open="[" close="]"><mfrac><msup><mi>x</mi><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></msup><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></mfrac></mfenced><mn>0</mn><mn>1</mn></msubsup></math> <em><strong>A1</strong></em></p>
<p>leading to <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> <em><strong>AG</strong></em></p>
<p> </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>METHOD 1</strong></p>
<p>use of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Var</mtext><mfenced><mi>X</mi></mfenced><mo>=</mo><mtext>E</mtext><mfenced><msup><mi>X</mi><mn>2</mn></msup></mfenced><mo>-</mo><msup><mfenced open="[" close="]"><mrow><mtext>E</mtext><mfenced><mi>X</mi></mfenced></mrow></mfenced><mn>2</mn></msup></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Var</mtext><mfenced><mi>X</mi></mfenced><mo>=</mo><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><munderover><mo>∫</mo><mn>0</mn><mn>1</mn></munderover><msup><mi>x</mi><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></msup><mo>d</mo><mi>x</mi><mo>-</mo><msup><mfenced><mfrac><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></mfrac></mfenced><mn>2</mn></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><msubsup><mfenced open="[" close="]"><mrow><mfrac><mn>1</mn><mrow><mi>n</mi><mo>+</mo><mn>3</mn></mrow></mfrac><msup><mi>x</mi><mrow><mi>n</mi><mo>+</mo><mn>3</mn></mrow></msup></mrow></mfenced><mn>0</mn><mn>1</mn></msubsup><mo>-</mo><msup><mfenced><mfrac><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></mfrac></mfenced><mn>2</mn></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>n</mi><mo>+</mo><mn>3</mn></mrow></mfrac><mo>-</mo><msup><mfenced><mfrac><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></mfrac></mfenced><mn>2</mn></msup></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><msup><mfenced><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mn>2</mn></msup><mo>-</mo><msup><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mfenced><mrow><mi>n</mi><mo>+</mo><mn>3</mn></mrow></mfenced></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> <em><strong>M1</strong></em></p>
<p><br><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>n</mi><mo>+</mo><mn>4</mn><mo>-</mo><mfenced><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>n</mi><mo>+</mo><mn>3</mn></mrow></mfenced></mrow></mfenced></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> <em><strong>A1</strong></em></p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mfenced><mrow><msup><mi>n</mi><mn>3</mn></msup><mo>+</mo><mn>5</mn><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>8</mn><mi>n</mi><mo>+</mo><mn>4</mn></mrow></mfenced><mo>-</mo><mfenced><mrow><msup><mi>n</mi><mn>3</mn></msup><mo>+</mo><mn>5</mn><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>7</mn><mi>n</mi><mo>+</mo><mn>3</mn></mrow></mfenced></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> <em><strong>A1</strong></em></p>
<p><br><strong>THEN</strong></p>
<p>so <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> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>use of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Var</mtext><mfenced><mi>X</mi></mfenced><mo>=</mo><mtext>E</mtext><msup><mfenced><mrow><mi>X</mi><mo>-</mo><mtext>E</mtext><mfenced><mi>X</mi></mfenced></mrow></mfenced><mn>2</mn></msup></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Var</mtext><mfenced><mi>X</mi></mfenced><mo>=</mo><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><munderover><mo>∫</mo><mn>0</mn><mn>1</mn></munderover><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mfrac><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></mfrac></mrow></mfenced><mn>2</mn></msup><msup><mi>x</mi><mi>n</mi></msup><mo>d</mo><mi>x</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><msubsup><mfenced open="[" close="]"><mrow><mfrac><mn>1</mn><mrow><mi>n</mi><mo>+</mo><mn>3</mn></mrow></mfrac><msup><mi>x</mi><mrow><mi>n</mi><mo>+</mo><mn>3</mn></mrow></msup><mo>-</mo><mfrac><mrow><mn>2</mn><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><msup><mfenced><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mn>2</mn></msup></mfrac><msup><mi>x</mi><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></msup><mo>+</mo><mfrac><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><msup><mfenced><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mn>2</mn></msup></mfrac><msup><mi>x</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></mfenced><mn>0</mn><mn>1</mn></msubsup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>n</mi><mo>+</mo><mn>3</mn></mrow></mfrac><mo>-</mo><msup><mfenced><mfrac><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></mfrac></mfenced><mn>2</mn></msup></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><msup><mfenced><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mi>n</mi><mo>+</mo><mn>3</mn></mrow></mfenced></mrow></mfenced></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> <em><strong>M1</strong></em></p>
<p><br><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>n</mi><mo>+</mo><mn>4</mn><mo>-</mo><mfenced><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>n</mi><mo>+</mo><mn>3</mn></mrow></mfenced></mrow></mfenced></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> <em><strong>A1</strong></em></p>
<p><strong><br>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mfenced><mrow><msup><mi>n</mi><mn>3</mn></msup><mo>+</mo><mn>5</mn><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>8</mn><mi>n</mi><mo>+</mo><mn>4</mn></mrow></mfenced><mo>-</mo><mfenced><mrow><msup><mi>n</mi><mn>3</mn></msup><mo>+</mo><mn>5</mn><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>7</mn><mi>n</mi><mo>+</mo><mn>3</mn></mrow></mfenced></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> <em><strong>A1</strong></em></p>
<p><br><strong>THEN</strong></p>
<p>so <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> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[6 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 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>
<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="T \sim N(196,{\text{ }}{24^2})">
<mi>T</mi>
<mo>∼</mo>
<mi>N</mi>
<mo stretchy="false">(</mo>
<mn>196</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<msup>
<mn>24</mn>
<mn>2</mn>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(T < 180) = 0.252">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>T</mi>
<mo><</mo>
<mn>180</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0.252</mn>
</math></span> <strong><em>(M1)A1</em></strong></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{P}}(T < {T_1}) = 0.05">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>T</mi>
<mo><</mo>
<mrow>
<msub>
<mi>T</mi>
<mn>1</mn>
</msub>
</mrow>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0.05</mn>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{T_1} = 157">
<mrow>
<msub>
<mi>T</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>157</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="F \sim N(210,{\text{ }}{\sigma ^2})">
<mi>F</mi>
<mo>∼</mo>
<mi>N</mi>
<mo stretchy="false">(</mo>
<mn>210</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<msup>
<mi>σ</mi>
<mn>2</mn>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(F < 235) = 0.79">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>F</mi>
<mo><</mo>
<mn>235</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0.79</mn>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{235 - 210}}{\sigma } = 0.806421">
<mfrac>
<mrow>
<mn>235</mn>
<mo>−</mo>
<mn>210</mn>
</mrow>
<mi>σ</mi>
</mfrac>
<mo>=</mo>
<mn>0.806421</mn>
</math></span> or equivalent <strong><em>(M1)(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sigma = 31.0">
<mi>σ</mi>
<mo>=</mo>
<mn>31.0</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">c.</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>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<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="{\text{P}}(X = 3) = {(0.1)^3}"> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>=</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <mn>0.1</mn> <msup> <mo stretchy="false">)</mo> <mn>3</mn> </msup> </mrow> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 0.001"> <mo>=</mo> <mn>0.001</mn> </math></span> <strong><em>AG</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(X = 4) = {\text{P}}(VV\bar VV) + {\text{P}}(V\bar VVV) + {\text{P}}(\bar VVVV)"> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>=</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>V</mi> <mi>V</mi> <mrow> <mover> <mi>V</mi> <mo stretchy="false">¯</mo> </mover> </mrow> <mi>V</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>V</mi> <mrow> <mover> <mi>V</mi> <mo stretchy="false">¯</mo> </mover> </mrow> <mi>V</mi> <mi>V</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mrow> <mover> <mi>V</mi> <mo stretchy="false">¯</mo> </mover> </mrow> <mi>V</mi> <mi>V</mi> <mi>V</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=" = 3 \times {(0.1)^3} \times 0.9"> <mo>=</mo> <mn>3</mn> <mo>×</mo> <mrow> <mo stretchy="false">(</mo> <mn>0.1</mn> <msup> <mo stretchy="false">)</mo> <mn>3</mn> </msup> </mrow> <mo>×</mo> <mn>0.9</mn> </math></span> (or equivalent) <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 0.0027"> <mo>=</mo> <mn>0.0027</mn> </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><strong>METHOD 1</strong></p>
<p>attempting to form equations in <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> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{9 + 3a + b}}{{2000}} = \frac{1}{{1000}}{\text{ }}(3a + b = - 7)"> <mfrac> <mrow> <mn>9</mn> <mo>+</mo> <mn>3</mn> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mrow> <mrow> <mn>2000</mn> </mrow> </mfrac> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mn>1000</mn> </mrow> </mfrac> <mrow> <mtext> </mtext> </mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>=</mo> <mo>−</mo> <mn>7</mn> <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="\frac{{16 + 4a + b}}{{2000}} \times \frac{9}{{10}} = \frac{{27}}{{10\,000}}{\text{ }}(4a + b = - 10)"> <mfrac> <mrow> <mn>16</mn> <mo>+</mo> <mn>4</mn> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mrow> <mrow> <mn>2000</mn> </mrow> </mfrac> <mo>×</mo> <mfrac> <mn>9</mn> <mrow> <mn>10</mn> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>27</mn> </mrow> <mrow> <mn>10</mn> <mspace width="thinmathspace"></mspace> <mn>000</mn> </mrow> </mfrac> <mrow> <mtext> </mtext> </mrow> <mo stretchy="false">(</mo> <mn>4</mn> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>=</mo> <mo>−</mo> <mn>10</mn> <mo stretchy="false">)</mo> </math></span> <strong><em>A1</em></strong></p>
<p>attempting to solve simultaneously <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = - 3,{\text{ }}b = 2"> <mi>a</mi> <mo>=</mo> <mo>−</mo> <mn>3</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>b</mi> <mo>=</mo> <mn>2</mn> </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}}(X = n) = \left( {\begin{array}{*{20}{c}} {n - 1} \\ 2 \end{array}} \right) \times {0.1^3} \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> <mrow> <mo>(</mo> <mrow> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>×</mo> <mrow> <msup> <mn>0.1</mn> <mn>3</mn> </msup> </mrow> <mo>×</mo> <mrow> <msup> <mn>0.9</mn> <mrow> <mi>n</mi> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{(n - 1)(n - 2)}}{{2000}} \times {0.9^{n - 3}}"> <mo>=</mo> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> </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> <strong><em>(M1)A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{{n^2} - 3n + 2}}{{2000}} \times {0.9^{n - 3}}"> <mo>=</mo> <mfrac> <mrow> <mrow> <msup> <mi>n</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>3</mn> <mi>n</mi> <mo>+</mo> <mn>2</mn> </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> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = - 3,b = 2"> <mi>a</mi> <mo>=</mo> <mo>−</mo> <mn>3</mn> <mo>,</mo> <mi>b</mi> <mo>=</mo> <mn>2</mn> </math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Condone the absence of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{0.9^{n - 3}}"> <mrow> <msup> <mn>0.9</mn> <mrow> <mi>n</mi> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </math></span> in the determination of the values of <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>
<p> </p>
<p><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(X = n) = \frac{{{n^2} - 3n + 2}}{{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> <mn>3</mn> <mi>n</mi> <mo>+</mo> <mn>2</mn> </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> <strong><em>(M1)</em></strong></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(X = n) = \left( {\begin{array}{*{20}{c}} {n - 1} \\ 2 \end{array}} \right) \times {0.1^3} \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> <mrow> <mo>(</mo> <mrow> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>×</mo> <mrow> <msup> <mn>0.1</mn> <mn>3</mn> </msup> </mrow> <mo>×</mo> <mrow> <msup> <mn>0.9</mn> <mrow> <mi>n</mi> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </math></span> <strong><em>(M1)</em></strong></p>
<p><strong>THEN</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{(n - 1)(n - 2)}}{{2000}} \times {0.9^{n - 3}}"> <mo>=</mo> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> </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> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(X = n - 1) = \frac{{(n - 2)(n - 3)}}{{2000}} \times {0.9^{n - 4}}"> <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> <mo>=</mo> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2000</mn> </mrow> </mfrac> <mo>×</mo> <mrow> <msup> <mn>0.9</mn> <mrow> <mi>n</mi> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mrow> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{P}}(X = n)}}{{{\text{P}}(X = n - 1)}} = \frac{{(n - 1)(n - 2)}}{{(n - 2)(n - 3)}} \times 0.9"> <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> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> <mo>×</mo> <mn>0.9</mn> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{0.9(n - 1)}}{{n - 3}}"> <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> <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="\frac{{{\text{P}}(X = n)}}{{{\text{P}}(X = n - 1)}} = \frac{{\frac{{{n^2} - 3n + 2}}{{2000}} \times {{0.9}^{n - 3}}}}{{\frac{{{{(n - 1)}^2} - 3(n - 1) + 2}}{{2000}} \times {{0.9}^{n - 4}}}}"> <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> <mfrac> <mrow> <mrow> <msup> <mi>n</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>3</mn> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> <mrow> <mn>2000</mn> </mrow> </mfrac> <mo>×</mo> <mrow> <msup> <mrow> <mn>0.9</mn> </mrow> <mrow> <mi>n</mi> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </mrow> <mrow> <mfrac> <mrow> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>3</mn> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> </mrow> <mrow> <mn>2000</mn> </mrow> </mfrac> <mo>×</mo> <mrow> <msup> <mrow> <mn>0.9</mn> </mrow> <mrow> <mi>n</mi> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mrow> </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{{0.9({n^2} - 3n + 2)}}{{({n^2} - 5n + 6)}}"> <mo>=</mo> <mfrac> <mrow> <mn>0.9</mn> <mo stretchy="false">(</mo> <mrow> <msup> <mi>n</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>3</mn> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mrow> <msup> <mi>n</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>5</mn> <mi>n</mi> <mo>+</mo> <mn>6</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </math></span> <strong><em>A1A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>A1 </em></strong>for a correct numerator and <strong><em>A1 </em></strong>for a correct denominator.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{0.9(n - 1)(n - 2)}}{{(n - 2)(n - 3)}}"> <mo>=</mo> <mfrac> <mrow> <mn>0.9</mn> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{0.9(n - 1)}}{{n - 3}}"> <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> <strong><em>AG</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(i) attempting to solve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{0.9(n - 1)}}{{n - 3}} = 1"> <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> <mo>=</mo> <mn>1</mn> </math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n"> <mi>n</mi> </math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 21"> <mi>n</mi> <mo>=</mo> <mn>21</mn> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{0.9(n - 1)}}{{n - 3}} < 1 \Rightarrow n > 21"> <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> <mo><</mo> <mn>1</mn> <mo stretchy="false">⇒</mo> <mi>n</mi> <mo>></mo> <mn>21</mn> </math></span> <strong><em>R1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{0.9(n - 1)}}{{n - 3}} > 1 \Rightarrow n < 21"> <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> <mo>></mo> <mn>1</mn> <mo stretchy="false">⇒</mo> <mi>n</mi> <mo><</mo> <mn>21</mn> </math></span> <strong><em>R1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X"> <mi>X</mi> </math></span> has two modes <strong><em>AG</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>R1R1 </em></strong>for a clearly labelled graphical representation of the two inequalities (using <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{P}}(X = n)}}{{{\text{P}}(X = n - 1)}}"> <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> </math></span>).</p>
<p> </p>
<p>(ii) the modes are 20 and 21 <strong><em>A1</em></strong></p>
<p><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">d.</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="Y \sim {\text{B}}(x,{\text{ }}0.1)"> <mi>Y</mi> <mo>∼</mo> <mrow> <mtext>B</mtext> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>0.1</mn> <mo stretchy="false">)</mo> </math></span> <strong><em>(A1)</em></strong></p>
<p>attempting to solve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(Y \geqslant 3) > 0.5"> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>⩾</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>></mo> <mn>0.5</mn> </math></span> (or equivalent <em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 - {\text{P}}(Y \leqslant 2) > 0.5"> <mn>1</mn> <mo>−</mo> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>⩽</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>></mo> <mn>0.5</mn> </math></span>) for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(M1) </em></strong>for attempting to solve an equality (obtaining <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 26.4"> <mi>x</mi> <mo>=</mo> <mn>26.4</mn> </math></span>).</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 27"> <mi>x</mi> <mo>=</mo> <mn>27</mn> </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="\sum\limits_{n = 0}^x {{\text{P}}(X = n) > 0.5} "> <munderover> <mo movablelimits="false">∑</mo> <mrow> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>x</mi> </munderover> <mrow> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>=</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>></mo> <mn>0.5</mn> </mrow> </math></span> <strong><em>(A1)</em></strong></p>
<p>attempting to solve for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <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="x = 27"> <mi>x</mi> <mo>=</mo> <mn>27</mn> </math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">e.</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>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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,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"></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the probability that Iqbal passes at least two of the papers he attempts.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that Iqbal passes his third paper, given that he passed only one previous paper.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<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><em><img src="data:image/png;base64,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"> <strong>A1A1A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for each correct column of probabilities.</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>probability (at least twice) =</p>
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {0.6 \times 0.7 \times 0.8} \right) + \left( {0.6 \times 0.7 \times 0.2} \right) + \left( {0.6 \times 0.3 \times 0.6} \right) + \left( {0.4 \times 0.6 \times 0.7} \right)"> <mrow> <mo>(</mo> <mrow> <mn>0.6</mn> <mo>×</mo> <mn>0.7</mn> <mo>×</mo> <mn>0.8</mn> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mn>0.6</mn> <mo>×</mo> <mn>0.7</mn> <mo>×</mo> <mn>0.2</mn> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mn>0.6</mn> <mo>×</mo> <mn>0.3</mn> <mo>×</mo> <mn>0.6</mn> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mn>0.4</mn> <mo>×</mo> <mn>0.6</mn> <mo>×</mo> <mn>0.7</mn> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>(M1)</strong></em></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {0.6 \times 0.7} \right) + \left( {0.6 \times 0.3 \times 0.6} \right) + \left( {0.4 \times 0.6 \times 0.7} \right)"> <mrow> <mo>(</mo> <mrow> <mn>0.6</mn> <mo>×</mo> <mn>0.7</mn> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mn>0.6</mn> <mo>×</mo> <mn>0.3</mn> <mo>×</mo> <mn>0.6</mn> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mn>0.4</mn> <mo>×</mo> <mn>0.6</mn> <mo>×</mo> <mn>0.7</mn> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award<em><strong> M1</strong></em> for summing all required probabilities.</p>
<p><strong>THEN</strong></p>
<p>= 0.696<em> <strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>P(passes third paper given only one paper passed before)</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{{\text{P}}\,\left( {{\text{passes third AND only one paper passed before}}} \right)}}{{{\text{P}}\,\left( {{\text{passes once in first two papers}}} \right)}}"> <mo>=</mo> <mfrac> <mrow> <mrow> <mtext>P</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>passes third AND only one paper passed before</mtext> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mrow> <mtext>P</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>passes once in first two papers</mtext> </mrow> </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{{\left( {0.6 \times 0.3 \times 0.6} \right) + \left( {0.4 \times 0.6 \times 0.7} \right)}}{{\left( {0.6 \times 0.3} \right) + \left( {0.4 \times 0.6} \right)}}"> <mo>=</mo> <mfrac> <mrow> <mrow> <mo>(</mo> <mrow> <mn>0.6</mn> <mo>×</mo> <mn>0.3</mn> <mo>×</mo> <mn>0.6</mn> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mn>0.4</mn> <mo>×</mo> <mn>0.6</mn> <mo>×</mo> <mn>0.7</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mrow> <mo>(</mo> <mrow> <mn>0.6</mn> <mo>×</mo> <mn>0.3</mn> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mn>0.4</mn> <mo>×</mo> <mn>0.6</mn> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p>= 0.657<em> <strong>A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</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>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<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="{\text{P}}(X < 42.52) = 0.6940">
<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>0.6940</mn>
</math></span> <strong><em>(M1)</em></strong></p>
<p>either <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {Z < \frac{{30.31 - \mu }}{\sigma }} \right) = 0.1180{\text{ or P}}\left( {Z < \frac{{42.52 - \mu }}{\sigma }} \right) = 0.6940">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>Z</mi>
<mo><</mo>
<mfrac>
<mrow>
<mn>30.31</mn>
<mo>−</mo>
<mi>μ</mi>
</mrow>
<mi>σ</mi>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0.1180</mn>
<mrow>
<mtext> or P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>Z</mi>
<mo><</mo>
<mfrac>
<mrow>
<mn>42.52</mn>
<mo>−</mo>
<mi>μ</mi>
</mrow>
<mi>σ</mi>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0.6940</mn>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{30.31 - \mu }}{\sigma } = \underbrace {{\Phi ^{ - 1}}(0.1180)}_{ - 1.1850 \ldots }">
<mfrac>
<mrow>
<mn>30.31</mn>
<mo>−</mo>
<mi>μ</mi>
</mrow>
<mi>σ</mi>
</mfrac>
<mo>=</mo>
<munder>
<mrow>
<munder>
<mrow>
<mrow>
<msup>
<mi mathvariant="normal">Φ</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo stretchy="false">(</mo>
<mn>0.1180</mn>
<mo stretchy="false">)</mo>
</mrow>
<mo>⏟</mo>
</munder>
</mrow>
<mrow>
<mo>−</mo>
<mn>1.1850</mn>
<mo>…</mo>
</mrow>
</munder>
</math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{42.52 - \mu }}{\sigma } = \underbrace {{\Phi ^{ - 1}}(0.6940)}_{0.5072 \ldots }">
<mfrac>
<mrow>
<mn>42.52</mn>
<mo>−</mo>
<mi>μ</mi>
</mrow>
<mi>σ</mi>
</mfrac>
<mo>=</mo>
<munder>
<mrow>
<munder>
<mrow>
<mrow>
<msup>
<mi mathvariant="normal">Φ</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo stretchy="false">(</mo>
<mn>0.6940</mn>
<mo stretchy="false">)</mo>
</mrow>
<mo>⏟</mo>
</munder>
</mrow>
<mrow>
<mn>0.5072</mn>
<mo>…</mo>
</mrow>
</munder>
</math></span> <strong><em>(A1)</em></strong></p>
<p>attempting to solve simultaneously <strong><em>(M1) </em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu = 38.9">
<mi>μ</mi>
<mo>=</mo>
<mn>38.9</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sigma = 7.22">
<mi>σ</mi>
<mo>=</mo>
<mn>7.22</mn>
</math></span> <strong><em>A1</em></strong></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="{\text{P}}(\mu - 1.2\sigma < X < \mu + 1.2\sigma )">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>μ</mi>
<mo>−</mo>
<mn>1.2</mn>
<mi>σ</mi>
<mo><</mo>
<mi>X</mi>
<mo><</mo>
<mi>μ</mi>
<mo>+</mo>
<mn>1.2</mn>
<mi>σ</mi>
<mo stretchy="false">)</mo>
</math></span> (or equivalent <em>eg</em>. <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2{\text{P}}(\mu < X < \mu + 1.2\sigma )">
<mn>2</mn>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>μ</mi>
<mo><</mo>
<mi>X</mi>
<mo><</mo>
<mi>μ</mi>
<mo>+</mo>
<mn>1.2</mn>
<mi>σ</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=" = 0.770">
<mo>=</mo>
<mn>0.770</mn>
</math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(M1)A1 </em></strong>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}( - 1.2 < Z < 1.2) = 0.770">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mo>−</mo>
<mn>1.2</mn>
<mo><</mo>
<mi>Z</mi>
<mo><</mo>
<mn>1.2</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0.770</mn>
</math></span>.</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>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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>use of formula or Venn diagram <em><strong> (M1)</strong></em></p>
<p>0.72 + 0.45 − 1 <em><strong>(A1)</strong></em></p>
<p>= 0.17 <em><strong>A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>0.72 − 0.17 = 0.55 <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>200 × 0.45 = 90 <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>let <em>X</em> be the number of customers who order cake</p>
<p><em>X</em> ~ B(200,0.45) <em><strong>(M1)</strong></em></p>
<p>P(<em>X</em> > 100) = P(<em>X</em> ≥ 101)(= 1 − P(<em>X</em> ≤ 100)) <em><strong>(M1)</strong></em></p>
<p>= 0.0681 <em><strong>A1</strong></em></p>
<p> </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>0.46 × 0.8 = 0.368 <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.i.</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="0.368 + 0.54 \times {\text{P}}\left( {S\left| F \right.} \right) = 0.72"> <mn>0.368</mn> <mo>+</mo> <mn>0.54</mn> <mo>×</mo> <mrow> <mtext>P</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>S</mi> <mrow> <mo>|</mo> <mi>F</mi> <mo symmetric="true" stretchy="true" fence="true"></mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0.72</mn> </math></span> <em><strong>M1</strong></em><em><strong>A1</strong></em><em><strong>A1</strong></em> </p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for an appropriate tree diagram. Award <em><strong>M1</strong></em> for LHS, <em><strong>M1</strong></em> for RHS.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {S\left| F \right.} \right) = 0.652"> <mrow> <mtext>P</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>S</mi> <mrow> <mo>|</mo> <mi>F</mi> <mo symmetric="true" stretchy="true" fence="true"></mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0.652</mn> </math></span> <em><strong>A1</strong></em> </p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {S\left| F \right.} \right) = \frac{{{\text{P}}\left( {S \cap F} \right)}}{{{\text{P}}\left( F \right)}}"> <mrow> <mtext>P</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>S</mi> <mrow> <mo>|</mo> <mi>F</mi> <mo symmetric="true" stretchy="true" fence="true"></mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mrow> <mtext>P</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>S</mi> <mo>∩</mo> <mi>F</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mrow> <mtext>P</mtext> </mrow> <mrow> <mo>(</mo> <mi>F</mi> <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{{0.72 - 0.368}}{{0.54}}"> <mo>=</mo> <mfrac> <mrow> <mn>0.72</mn> <mo>−</mo> <mn>0.368</mn> </mrow> <mrow> <mn>0.54</mn> </mrow> </mfrac> </math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong> </em>for numerator, <em><strong>A1</strong> </em>for denominator.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {S\left| F \right.} \right) = 0.652"> <mrow> <mtext>P</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>S</mi> <mrow> <mo>|</mo> <mi>F</mi> <mo symmetric="true" stretchy="true" fence="true"></mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0.652</mn> </math></span> <em><strong>A1</strong></em> </p>
<p> </p>
<p><em><strong>[4 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.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</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>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>
<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="{\text{P}}(X < 250) = 0.0228">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo><</mo>
<mn>250</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0.0228</mn>
</math></span> <strong><em>(M1)A1</em></strong></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="\frac{{250 - \mu }}{{1.5}} = - 2.878 \ldots ">
<mfrac>
<mrow>
<mn>250</mn>
<mo>−</mo>
<mi>μ</mi>
</mrow>
<mrow>
<mn>1.5</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mo>−</mo>
<mn>2.878</mn>
<mo>…</mo>
</math></span> <strong><em>(M1)(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \mu = 254.32">
<mo stretchy="false">⇒</mo>
<mi>μ</mi>
<mo>=</mo>
<mn>254.32</mn>
</math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Notes:</strong> Only award <strong><em>A1 </em></strong>here if the correct 2dp answer is seen. Award <strong><em>M0 </em></strong>for use of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{1.5^2}">
<mrow>
<msup>
<mn>1.5</mn>
<mn>2</mn>
</msup>
</mrow>
</math></span>.</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{250 - 253}}{\sigma } = - 2.878 \ldots ">
<mfrac>
<mrow>
<mn>250</mn>
<mo>−</mo>
<mn>253</mn>
</mrow>
<mi>σ</mi>
</mfrac>
<mo>=</mo>
<mo>−</mo>
<mn>2.878</mn>
<mo>…</mo>
</math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \sigma = 1.04">
<mo stretchy="false">⇒</mo>
<mi>σ</mi>
<mo>=</mo>
<mn>1.04</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">c.</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>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>recognition of the need to integrate <math xmlns="http://www.w3.org/1998/Math/MathML"><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></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><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>d</mo><mi>x</mi><mfenced><mrow><mo>=</mo><mn>1</mn></mrow></mfenced></math></p>
<p> </p>
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>k</mi><mo>⇒</mo><mfrac><mrow><mo>d</mo><mi>u</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mn>2</mn><mi>x</mi></math> (or equivalent) <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><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>d</mo><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>∫</mo><msup><mi>u</mi><mrow><mo>-</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></msup><mo>d</mo><mi>u</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><msup><mi>u</mi><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mfenced><mrow><mo>+</mo><mi>c</mi></mrow></mfenced><mo> </mo><mfenced><mrow><mo>=</mo><mo>-</mo><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>k</mi></mrow></mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mfenced><mrow><mo>+</mo><mi>c</mi></mrow></mfenced></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><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>d</mo><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>∫</mo><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><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>d</mo><mi>x</mi></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>k</mi></mrow></mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mfenced><mrow><mo>+</mo><mi>c</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>THEN</strong></p>
<p>attempt to use correct limits for their integrand and set equal 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"><msubsup><mfenced open="[" close="]"><mrow><mo>-</mo><msup><mi>u</mi><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup></mrow></mfenced><mi>k</mi><mrow><mn>16</mn><mo>+</mo><mi>k</mi></mrow></msubsup><mo>=</mo><mn>1</mn></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mfenced open="[" close="]"><mrow><mo>-</mo><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>k</mi></mrow></mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup></mrow></mfenced><mn>0</mn><mn>4</mn></msubsup><mo>=</mo><mn>1</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><msup><mfenced><mrow><mn>16</mn><mo>+</mo><mi>k</mi></mrow></mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mo>+</mo><msup><mi>k</mi><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mo>=</mo><mn>1</mn><mfenced><mrow><mo>⇒</mo><mfrac><mn>1</mn><msqrt><mi>k</mi></msqrt></mfrac><mo>-</mo><mfrac><mn>1</mn><msqrt><mn>16</mn><mo>+</mo><mi>k</mi></msqrt></mfrac><mo>=</mo><mn>1</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><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> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to solve <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> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>645038</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>645</mn></math> <em><strong>A1</strong></em></p>
<p> </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><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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><em>a</em> = 1.29 and <em>b</em> = −10.4 <em><strong> A1A1</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>recognising both lines pass through the mean point <em><strong> (M1)</strong></em></p>
<p><em>p</em> = 28.7, <em>q</em> = 30.3 <em><strong>A2</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>substitution into <strong>their</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> on <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span> equation <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> = 1.29082(29) − 10.3793</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> = 27.1 <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Accept 27.</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</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>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>use of GDC to give <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>883529</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>884</mn></math> <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Award the <em><strong>(M1)</strong></em> for any correct value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>r</mi><mn>2</mn></msup><mo>=</mo><mn>0</mn><mo>.</mo><mn>780624</mn><mo>…</mo></math> seen in part (a) or part (b).</p>
<p><br><em><strong>[2 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"><mi>a</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>36609</mn><mo>…</mo><mo> </mo><mo>,</mo><mo> </mo><mi>b</mi><mo>=</mo><mn>64</mn><mo>.</mo><mn>5171</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>37</mn><mo> </mo><mo>,</mo><mo> </mo><mi>b</mi><mo>=</mo><mn>64</mn><mo>.</mo><mn>5</mn></math> <em><strong>A1</strong></em></p>
<p><strong><br></strong><br><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to find their difference <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>×</mo><mn>1</mn><mo>.</mo><mn>36609</mn><mo>…</mo></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>36609</mn><mo>…</mo><mfenced><mrow><mi>h</mi><mo>+</mo><mn>5</mn></mrow></mfenced><mo>+</mo><mn>64</mn><mo>.</mo><mn>5171</mn><mo>…</mo><mo>-</mo><mfenced><mrow><mn>1</mn><mo>.</mo><mn>36609</mn><mo>…</mo><mi>h</mi><mo>+</mo><mn>64</mn><mo>.</mo><mn>5171</mn><mo>…</mo></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>.</mo><mn>83045</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>6</mn><mo>.</mo><mn>83</mn></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>6</mn><mo>.</mo><mn>85</mn><mo> </mo><mi>from</mi><mo> </mo><mn>1</mn><mo>.</mo><mn>37</mn></mrow></mfenced></math></p>
<p>the student could have expected her score to increase by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn></math> marks. <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Accept an increase of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>,</mo><mo> </mo><mn>6</mn><mo>.</mo><mn>83</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>.</mo><mn>85</mn></math>.</p>
<p><br><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Lucy is incorrect in suggesting there is a causal relationship.</p>
<p>This might be true, but the data can only indicate a correlation. <em><strong>R1</strong></em></p>
<p><br><strong>Note:</strong> Accept ‘Lucy is incorrect as correlation does not imply causation’ or equivalent.</p>
<p><br><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>no effect <em><strong>A1</strong></em></p>
<p><br><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">e.</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>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><em><strong>METHOD 1</strong></em></p>
<p>attempting to use the expected value formula<em><strong> (M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{E}}\left( X \right) = \left( {1 \times 0.60} \right) + \left( {2 \times 0.30} \right) + \left( {3 \times 0.03} \right) + \left( {4 \times 0.05} \right) + \left( {5 \times 0.02} \right)">
<mrow>
<mtext>E</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mi>X</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>×</mo>
<mn>0.60</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mo>×</mo>
<mn>0.30</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>3</mn>
<mo>×</mo>
<mn>0.03</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>4</mn>
<mo>×</mo>
<mn>0.05</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>5</mn>
<mo>×</mo>
<mn>0.02</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{E}}\left( X \right) = 1.59">
<mrow>
<mtext>E</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mi>X</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>1.59</mn>
</math></span>($) <em><strong>(A1)</strong></em></p>
<p>use of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{E}}\left( {1.20X + 2.40} \right) = 1.20{\text{E}}\left( X \right) + 2.40">
<mrow>
<mtext>E</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>1.20</mn>
<mi>X</mi>
<mo>+</mo>
<mn>2.40</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>1.20</mn>
<mrow>
<mtext>E</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mi>X</mi>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mn>2.40</mn>
</math></span><em><strong> (M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{E}}\left( T \right) = 1.20\left( {1.59} \right) + 2.40">
<mrow>
<mtext>E</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mi>T</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>1.20</mn>
<mrow>
<mo>(</mo>
<mrow>
<mn>1.59</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mn>2.40</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 4.31">
<mo>=</mo>
<mn>4.31</mn>
</math></span>($)<em><strong> A1</strong></em></p>
<p> </p>
<p><em><strong>METHOD 2</strong></em></p>
<p>attempting to find the probability distribution for <em>T</em> <em><strong>(M1)</strong></em></p>
<p><img src="data:image/png;base64,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"> <em><strong>(A1)</strong></em></p>
<p>attempting to use the expected value formula <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{E}}\left( T \right) = \left( {3.60 \times 0.60} \right) + \left( {4.80 \times 0.30} \right) + \left( {6.00 \times 0.03} \right) + \left( {7.20 \times 0.05} \right) + \left( {8.40 \times 0.02} \right)">
<mrow>
<mtext>E</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mi>T</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>3.60</mn>
<mo>×</mo>
<mn>0.60</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>4.80</mn>
<mo>×</mo>
<mn>0.30</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>6.00</mn>
<mo>×</mo>
<mn>0.03</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>7.20</mn>
<mo>×</mo>
<mn>0.05</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>8.40</mn>
<mo>×</mo>
<mn>0.02</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 4.31">
<mo>=</mo>
<mn>4.31</mn>
</math></span>($)<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><em><strong>METHOD 1</strong></em></p>
<p>using Var(1.20 <em>X</em> + 2.40) = (1.20)<sup>2</sup> Var(<em>X</em>) with Var(<em>X</em>) = 0.8419 <em><strong>(M1)</strong></em></p>
<p>Var(<em>T</em>) = 1.21<em><strong> A1</strong></em></p>
<p><em><strong>METHOD 2</strong></em></p>
<p>finding the standard deviation for <strong>their</strong> probability distribution found in part (a) <em><strong>(M1)</strong></em></p>
<p>Var(<em>T</em>) = (1.101…)<sup>2</sup> </p>
<p>= 1.21<em><strong> A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1A1</strong> </em>for Var(<em>T</em>) = (1.093…)<sup>2</sup> = 1.20.</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><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>
<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><em>X</em> is number of squirrels in reserve<br><em>X</em> ∼ Po(179.2) <em><strong> A1</strong></em></p>
<p><strong>Note:</strong> Award<em><strong> A1</strong></em> if 179.2 or 56 × 3.2 seen or implicit in future calculations.</p>
<p>recognising conditional probability <em><strong>M1</strong></em></p>
<p>P(<em>X</em> > 190 | <em>X</em> ≥ 168)</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{{\text{P}}\left( {X > 190} \right)}}{{{\text{P}}\left( {X \geqslant 168} \right)}} = \left( {\frac{{0.19827 \ldots }}{{0.80817 \ldots }}} \right)">
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo>></mo>
<mn>190</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo>⩾</mo>
<mn>168</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>0.19827</mn>
<mo>…</mo>
</mrow>
<mrow>
<mn>0.80817</mn>
<mo>…</mo>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(A1)(A1)</strong></em></p>
<p>= 0.245 <em><strong>A1</strong></em></p>
<p><em><strong>[5 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>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>
<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>P(<em>L</em> ≥ 5) = 0.910 <em><strong>(M1)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>X</em> is the number of wolves found to be at least 5 years old recognising binomial distribution <em><strong>M1</strong></em></p>
<p><em>X</em> ~ B(8, 0.910…)</p>
<p>P(<em>X</em> > 6) = 1 − P(<em>X</em> ≤ 6) <em><strong>(M1)</strong></em></p>
<p>= 0.843 <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1A0</strong></em> for finding P(<em>X</em> ≥ 6).</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 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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color:#999;font-size:90%;font-style:italic;">* This sample question was produced by experienced DP mathematics senior examiners to aid teachers in preparing for external assessment in the new MAA course. There may be minor differences in formatting compared to formal exam papers.</p>
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mi>n</mi></msub></math> is the event “the first tail occurs on the <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math>nd, <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math>th, <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math>th, …, <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>n</mi></math>th toss”</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mi>E</mi></mfenced><mo>=</mo><munderover><mtext>Σ</mtext><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mo>∞</mo></munderover><mtext>P</mtext><mfenced><msub><mi>E</mi><mi>n</mi></msub></mfenced></math> <strong>(A1)</strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong>A1</strong> for deducing that either <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> head before a tail or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> heads before a tail or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math> heads before a tail etc. is required. In other words, deduces <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>2</mn><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced></math> heads before a tail.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mi>E</mi></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>4</mn><mo>×</mo><mn>0</mn><mo>.</mo><mn>6</mn><mo>+</mo><msup><mfenced><mrow><mn>0</mn><mo>.</mo><mn>4</mn></mrow></mfenced><mn>3</mn></msup><mo>×</mo><mn>0</mn><mo>.</mo><mn>6</mn><mo>+</mo><msup><mfenced><mrow><mn>0</mn><mo>.</mo><mn>4</mn></mrow></mfenced><mn>5</mn></msup><mo>×</mo><mn>0</mn><mo>.</mo><mn>6</mn><mo>+</mo><mo>…</mo></math> <strong>M1A1</strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong>M1</strong> for attempting to form an infinite geometric series.</p>
<p><strong>Note:</strong> Award <strong>A1</strong> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mi>E</mi></mfenced><mo>=</mo><munderover><mtext>Σ</mtext><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mo>∞</mo></munderover><msup><mfenced><mrow><mn>0</mn><mo>.</mo><mn>4</mn></mrow></mfenced><mrow><mn>2</mn><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mn>0</mn><mo>.</mo><mn>6</mn></mrow></mfenced></math>.</p>
<p> </p>
<p>uses <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mo>∞</mo></msub><mo>=</mo><mfrac><msub><mi>u</mi><mn>1</mn></msub><mrow><mn>1</mn><mo>-</mo><mi>r</mi></mrow></mfrac></math> with <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub><mo>=</mo><mn>0</mn><mo>.</mo><mn>6</mn><mo>×</mo><mn>0</mn><mo>.</mo><mn>4</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><msup><mfenced><mrow><mn>0</mn><mo>.</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup></math> <strong>(M1)</strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong>M1</strong> for using <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mo>∞</mo></msub><mo>=</mo><mfrac><msub><mi>u</mi><mn>1</mn></msub><mrow><mn>1</mn><mo>-</mo><mi>r</mi></mrow></mfrac></math> with <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub><mo>=</mo><mn>0</mn><mo>.</mo><mn>4</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><msup><mfenced><mrow><mn>0</mn><mo>.</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup></math></p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>0</mn><mo>.</mo><mn>6</mn><mo>×</mo><mn>0</mn><mo>.</mo><mn>4</mn></mrow><mrow><mn>1</mn><mo>-</mo><msup><mfenced><mrow><mn>0</mn><mo>.</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup></mrow></mfrac></math> <strong>A1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>286</mn><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mn>2</mn><mn>7</mn></mfrac></mrow></mfenced></math> <strong>A1</strong></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>let <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>1</mn></msub></math> be the event “tail occurs on the first toss”</p>
<p>uses <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mi>E</mi></mfenced><mo>=</mo><mtext>P</mtext><mfenced><mrow><mi>E</mi><mo> </mo><menclose notation="left"><msub><mi>T</mi><mn>1</mn></msub></menclose></mrow></mfenced><mtext>P</mtext><mfenced><msub><mi>T</mi><mn>1</mn></msub></mfenced><mo>+</mo><mtext>P</mtext><mfenced><mrow><mi>E</mi><mo> </mo><menclose notation="left"><msub><mi>T</mi><mn>1</mn></msub><mspace width="-0.2em"></mspace><mo>'</mo></menclose></mrow></mfenced><mtext>P</mtext><mfenced><mrow><msub><mi>T</mi><mn>1</mn></msub><mspace width="-0.2em"></mspace><mo>'</mo></mrow></mfenced></math> <strong>M1</strong></p>
<p>concludes that <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>E</mi><mo> </mo><menclose notation="left"><msub><mi>T</mi><mn>1</mn></msub></menclose></mrow></mfenced><mo>=</mo><mn>0</mn></math> and so <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mi>E</mi></mfenced><mo>=</mo><mtext>P</mtext><mfenced><mrow><mi>E</mi><mo> </mo><menclose notation="left"><msub><mi>T</mi><mn>1</mn></msub><mspace width="-0.2em"></mspace><mo>'</mo></menclose></mrow></mfenced><mtext>P</mtext><mfenced><mrow><msub><mi>T</mi><mn>1</mn></msub><mspace width="-0.2em"></mspace><mo>'</mo></mrow></mfenced></math> <strong>R1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>E</mi><mo> </mo><menclose notation="left"><msub><mi>T</mi><mn>1</mn></msub><mspace width="-0.2em"></mspace><mo>'</mo></menclose></mrow></mfenced><mo>=</mo><mtext>P</mtext><mfenced><mrow><mi>E</mi><mo>'</mo></mrow></mfenced><mfenced><mrow><mo>=</mo><mn>1</mn><mo>-</mo><mtext>P</mtext><mfenced><mi>E</mi></mfenced></mrow></mfenced></math> <strong>A1</strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong>A1</strong> for concluding: given that a tail is not obtained on the first toss, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>E</mi><mo> </mo><menclose notation="left"><msub><mi>T</mi><mn>1</mn></msub><mspace width="-0.2em"></mspace><mo>'</mo></menclose></mrow></mfenced></math> is the probability that the first tail is obtained after a further odd number of tosses, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>E</mi><mo>'</mo></mrow></mfenced></math>.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><msub><mi>T</mi><mn>1</mn></msub><mspace width="-0.2em"></mspace><mo>'</mo></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.4</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mi>E</mi></mfenced><mo>=</mo><mn>0</mn><mo>.4</mo><mfenced><mrow><mn>1</mn><mo>-</mo><mtext>P</mtext><mfenced><mi>E</mi></mfenced></mrow></mfenced></math> <strong>A1</strong></p>
<p>attempts to solve for <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mi>E</mi></mfenced></math> <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>286</mn><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mn>2</mn><mn>7</mn></mfrac></mrow></mfenced></math> <strong>A1</strong></p>
<p> </p>
<p><strong>[6 marks]</strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</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>
<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><em>np</em> = 3.5 <em><strong>(A1)</strong></em></p>
<p><em>p </em>≤ 1 ⇒ least <em>n</em> = 4 <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>(1 − <em>p</em>)<em><sup>n</sup></em> + <em>np</em>(1 − <em>p</em>)<sup><em>n</em>−1</sup> = 0.09478 <em><strong>M1A1</strong></em></p>
<p>attempt to solve above equation with <em>np</em> = 3.5 <em><strong>(M1)</strong></em></p>
<p><em>n</em> = 12, <em>p</em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{7}{{24}}">
<mfrac>
<mn>7</mn>
<mrow>
<mn>24</mn>
</mrow>
</mfrac>
</math></span> (=0.292) <em><strong>A1A1</strong></em></p>
<p><strong>Note:</strong> Do not accept <em>n</em> as a decimal.</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 <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>
<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> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a\left[ {\int_0^{0.5} {3x\,{\text{d}}x} + \int_{0.5}^2 {\left( {2 - x} \right)} \,{\text{d}}x} \right] = 1"> <mi>a</mi> <mrow> <mo>[</mo> <mrow> <msubsup> <mo>∫</mo> <mn>0</mn> <mrow> <mn>0.5</mn> </mrow> </msubsup> <mrow> <mn>3</mn> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> <mo>+</mo> <msubsup> <mo>∫</mo> <mrow> <mn>0.5</mn> </mrow> <mn>2</mn> </msubsup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mo>−</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mn>1</mn> </math></span> <em><strong>M1</strong></em></p>
<p><strong>Note</strong>: Award the <em><strong>M1</strong></em> for the total integral equalling 1, or equivalent.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a\left( {\frac{3}{2}} \right) = 1"> <mi>a</mi> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>1</mn> </math></span> <em><strong>(M1)A1</strong></em></p>
<p><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> <em><strong>AG</strong></em></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>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^{0.5} {2x\,{\text{d}}x} + \frac{2}{3}\int_{0.5}^1 {\left( {2 - x} \right)} \,{\text{d}}x"> <msubsup> <mo>∫</mo> <mn>0</mn> <mrow> <mn>0.5</mn> </mrow> </msubsup> <mrow> <mn>2</mn> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> <mo>+</mo> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> <msubsup> <mo>∫</mo> <mrow> <mn>0.5</mn> </mrow> <mn>1</mn> </msubsup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mo>−</mo> <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)(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{2}{3}"> <mo>=</mo> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </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="\frac{2}{3}\int_1^2 {\left( {2 - x} \right)} \,{\text{d}}x = \frac{1}{3}"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> <msubsup> <mo>∫</mo> <mn>1</mn> <mn>2</mn> </msubsup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mo>−</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </math></span> <em><strong>(M1)</strong></em></p>
<p>so <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {X < 1} \right) = \frac{2}{3}"> <mrow> <mtext>P</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>X</mi> <mo><</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </math></span> <em><strong>(M1)A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {s < X < 0.8} \right) = \int_s^{0.5} {2x\,{\text{d}}x} + \frac{2}{3}\int_{0.5}^{0.8} {\left( {2 - x} \right)} \,{\text{d}}x"> <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> <msubsup> <mo>∫</mo> <mi>s</mi> <mrow> <mn>0.5</mn> </mrow> </msubsup> <mrow> <mn>2</mn> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> <mo>+</mo> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> <msubsup> <mo>∫</mo> <mrow> <mn>0.5</mn> </mrow> <mrow> <mn>0.8</mn> </mrow> </msubsup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mo>−</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </math></span> <em><strong>M1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left[ {{x^2}} \right]_s^{0.5} + 0.27"> <mo>=</mo> <msubsup> <mrow> <mo>[</mo> <mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> <mo>]</mo> </mrow> <mi>s</mi> <mrow> <mn>0.5</mn> </mrow> </msubsup> <mo>+</mo> <mn>0.27</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0.25 - {s^2} + 0.27"> <mn>0.25</mn> <mo>−</mo> <mrow> <msup> <mi>s</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>0.27</mn> </math></span> <strong> <em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {2s < X < 0.8} \right) = \frac{2}{3}\int_{2s}^{0.8} {\left( {2 - x} \right)} \,{\text{d}}x"> <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> <mo>=</mo> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> <msubsup> <mo>∫</mo> <mrow> <mn>2</mn> <mi>s</mi> </mrow> <mrow> <mn>0.8</mn> </mrow> </msubsup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mo>−</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{2}{3}\left[ {2x - \frac{{{x^2}}}{2}} \right]_{2s}^{0.8}"> <mo>=</mo> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> <msubsup> <mrow> <mo>[</mo> <mrow> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mfrac> <mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>]</mo> </mrow> <mrow> <mn>2</mn> <mi>s</mi> </mrow> <mrow> <mn>0.8</mn> </mrow> </msubsup> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{2}{3}\left( {1.28 - \left( {4s - 2{s^2}} \right)} \right)"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> <mrow> <mo>(</mo> <mrow> <mn>1.28</mn> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mn>4</mn> <mi>s</mi> <mo>−</mo> <mn>2</mn> <mrow> <msup> <mi>s</mi> <mn>2</mn> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p>equating</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0.25 - {s^2} + 0.27 = \frac{4}{3}\left( {1.28 - \left( {4s - 2{s^2}} \right)} \right)"> <mn>0.25</mn> <mo>−</mo> <mrow> <msup> <mi>s</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>0.27</mn> <mo>=</mo> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> <mrow> <mo>(</mo> <mrow> <mn>1.28</mn> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mn>4</mn> <mi>s</mi> <mo>−</mo> <mn>2</mn> <mrow> <msup> <mi>s</mi> <mn>2</mn> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>(A1)</strong></em></p>
<p>attempt to solve for <em>s</em> <em><strong>(M1)</strong></em></p>
<p><em>s</em> = 0.274 <em><strong>A1</strong></em></p>
<p><em><strong>[7 marks]</strong></em></p>
<div class="question_part_label">c.</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>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<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="\int\limits_0^4 {\left( {\frac{{{x^2}}}{a} + b} \right){\text{d}}x = 1 \Rightarrow \left[ {\frac{{{x^3}}}{{3a}} + bx} \right]_0^4 = 1 \Rightarrow \frac{{64}}{{3a}} + 4b = 1} "> <munderover> <mo>∫</mo> <mn>0</mn> <mn>4</mn> </munderover> <mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> <mi>a</mi> </mfrac> <mo>+</mo> <mi>b</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> <mn>1</mn> <mo stretchy="false">⇒</mo> <msubsup> <mrow> <mo>[</mo> <mrow> <mfrac> <mrow> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> </mrow> <mrow> <mn>3</mn> <mi>a</mi> </mrow> </mfrac> <mo>+</mo> <mi>b</mi> <mi>x</mi> </mrow> <mo>]</mo> </mrow> <mn>0</mn> <mn>4</mn> </msubsup> <mo>=</mo> <mn>1</mn> <mo stretchy="false">⇒</mo> <mfrac> <mrow> <mn>64</mn> </mrow> <mrow> <mn>3</mn> <mi>a</mi> </mrow> </mfrac> <mo>+</mo> <mn>4</mn> <mi>b</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int\limits_2^4 {\left( {\frac{{{x^2}}}{a} + b} \right){\text{d}}x = 0.75 \Rightarrow \frac{{56}}{{3a}} + 2b = 0.75} "> <munderover> <mo>∫</mo> <mn>2</mn> <mn>4</mn> </munderover> <mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> <mi>a</mi> </mfrac> <mo>+</mo> <mi>b</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> <mn>0.75</mn> <mo stretchy="false">⇒</mo> <mfrac> <mrow> <mn>56</mn> </mrow> <mrow> <mn>3</mn> <mi>a</mi> </mrow> </mfrac> <mo>+</mo> <mn>2</mn> <mi>b</mi> <mo>=</mo> <mn>0.75</mn> </mrow> </math></span> <strong><em>M1A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int\limits_0^2 {\left( {\frac{{{x^2}}}{a} + b} \right)} \,dx = 0.25 \Rightarrow \frac{8}{{3a}} + 2b = 0.25"> <munderover> <mo>∫</mo> <mn>0</mn> <mn>2</mn> </munderover> <mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> <mi>a</mi> </mfrac> <mo>+</mo> <mi>b</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mn>0.25</mn> <mo stretchy="false">⇒</mo> <mfrac> <mn>8</mn> <mrow> <mn>3</mn> <mi>a</mi> </mrow> </mfrac> <mo>+</mo> <mn>2</mn> <mi>b</mi> <mo>=</mo> <mn>0.25</mn> </math></span> could be seen/used in place of either of the above equations.</p>
<p> </p>
<p>evidence of an attempt to solve simultaneously (or check given <em>a</em>,<em>b </em>values are consistent) <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = 32,{\text{ }}b = \frac{1}{{12}}"> <mi>a</mi> <mo>=</mo> <mn>32</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>b</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mn>12</mn> </mrow> </mfrac> </math></span> <strong><em>AG</em></strong></p>
<p><strong><em>[5 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}}\left( X \right) = \int\limits_0^4 {x\left( {\frac{{{x^2}}}{{32}} + \frac{1}{{12}}} \right){\text{d}}x} "> <mrow> <mtext>E</mtext> </mrow> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mo>∫</mo> <mn>0</mn> <mn>4</mn> </munderover> <mrow> <mi>x</mi> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>32</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <mrow> <mn>12</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{E}}(X) = \frac{8}{3}\,\,\,( = 2.67)"> <mrow> <mtext>E</mtext> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mfrac> <mn>8</mn> <mn>3</mn> </mfrac> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mo stretchy="false">(</mo> <mo>=</mo> <mn>2.67</mn> <mo stretchy="false">)</mo> </math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</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^2}} \right) = \int\limits_0^4 {{x^2}\left( {\frac{{{x^2}}}{{32}} + \frac{1}{{12}}} \right){\text{d}}x} "> <mrow> <mtext>E</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mi>X</mi> <mn>2</mn> </msup> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mo>∫</mo> <mn>0</mn> <mn>4</mn> </munderover> <mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>32</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <mrow> <mn>12</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{Var}}(X) = {\text{E}}({X^2}) - {[{\text{E}}(X)]^2} = \frac{{16}}{{15}}\,\,\,( = 1.07)"> <mrow> <mtext>Var</mtext> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mtext>E</mtext> </mrow> <mo stretchy="false">(</mo> <mrow> <msup> <mi>X</mi> <mn>2</mn> </msup> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mrow> <mo stretchy="false">[</mo> <mrow> <mtext>E</mtext> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">]</mo> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mfrac> <mrow> <mn>16</mn> </mrow> <mrow> <mn>15</mn> </mrow> </mfrac> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mo stretchy="false">(</mo> <mo>=</mo> <mn>1.07</mn> <mo stretchy="false">)</mo> </math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">c.</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^m {\left( {\frac{{{x^2}}}{{32}} + \frac{1}{{12}}} \right){\text{d}}x = 0.5} "> <munderover> <mo>∫</mo> <mn>0</mn> <mi>m</mi> </munderover> <mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>32</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <mrow> <mn>12</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{m^3}}}{{96}} + \frac{m}{{12}} = 0.5\,\,\,( \Rightarrow {m^3} + 8m - 48 = 0)"> <mfrac> <mrow> <mrow> <msup> <mi>m</mi> <mn>3</mn> </msup> </mrow> </mrow> <mrow> <mn>96</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mi>m</mi> <mrow> <mn>12</mn> </mrow> </mfrac> <mo>=</mo> <mn>0.5</mn> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mo stretchy="false">(</mo> <mo stretchy="false">⇒</mo> <mrow> <msup> <mi>m</mi> <mn>3</mn> </msup> </mrow> <mo>+</mo> <mn>8</mn> <mi>m</mi> <mo>−</mo> <mn>48</mn> <mo>=</mo> <mn>0</mn> <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="m = 2.91"> <mi>m</mi> <mo>=</mo> <mn>2.91</mn> </math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="Y \sim B(8,{\text{ }}0.75)"> <mi>Y</mi> <mo>∼</mo> <mi>B</mi> <mo stretchy="false">(</mo> <mn>8</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>0.75</mn> <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{E}}(Y) = 8 \times 0.75 = 6"> <mrow> <mtext>E</mtext> </mrow> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>8</mn> <mo>×</mo> <mn>0.75</mn> <mo>=</mo> <mn>6</mn> </math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(Y \geqslant 3) = 0.996"> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>⩾</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.996</mn> </math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">f.</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>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>Rachel: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>~</mo><mtext>N</mtext><mfenced><mrow><mn>56</mn><mo>.</mo><mn>5</mn><mo>,</mo><msup><mn>3</mn><mn>2</mn></msup></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>R</mi><mo>≥</mo><mn>60</mn></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>1216</mn><mo>…</mo></math> <em><strong>(A1)</strong></em></p>
<p>Sophia: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mo>~</mo><mtext>N</mtext><mfenced><mrow><mn>57</mn><mo>.</mo><mn>5</mn><mo>,</mo><mn>1</mn><mo>.</mo><msup><mn>8</mn><mn>2</mn></msup></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>S</mi><mo>≥</mo><mn>60</mn></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>0824</mn><mo>…</mo></math> <em><strong>(A1)</strong></em></p>
<p>recognises binomial distribution with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>5</mn></math> <em><strong>(M1)</strong></em></p>
<p>let <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>N</mi><mi>R</mi></msub></math> represent the number of Rachel’s throws that are longer than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>60</mn></math> metres</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>N</mi><mi>R</mi></msub><mo>~</mo><mtext>B</mtext><mfenced><mrow><mn>5</mn><mo>,</mo><mn>0</mn><mo>.</mo><mn>1216</mn><mo>…</mo></mrow></mfenced></math></p>
<p>either <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><msub><mi>N</mi><mi>R</mi></msub><mo>≥</mo><mn>1</mn></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>4772</mn><mo>…</mo></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><msub><mi>N</mi><mi>R</mi></msub><mo>=</mo><mn>0</mn></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>5227</mn><mo>…</mo></math> <em><strong>(A1)</strong></em></p>
<p>let <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>N</mi><mi>S</mi></msub></math> represent the number of Sophia’s throws that are longer than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>60</mn></math> metres</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>N</mi><mi>S</mi></msub><mo>~</mo><mtext>B</mtext><mfenced><mrow><mn>5</mn><mo>,</mo><mn>0</mn><mo>.</mo><mn>0824</mn><mo>…</mo></mrow></mfenced></math></p>
<p>either <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><msub><mi>N</mi><mi>S</mi></msub><mo>≥</mo><mn>1</mn></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>3495</mn><mo>…</mo></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><msub><mi>N</mi><mi>S</mi></msub><mo>=</mo><mn>0</mn></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>6504</mn><mo>…</mo></math> <em><strong>(A1)</strong></em></p>
<p><strong><br>EITHER</strong></p>
<p>uses <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><msub><mi>N</mi><mi>R</mi></msub><mo>≥</mo><mn>1</mn></mrow></mfenced><mtext>P</mtext><mfenced><mrow><msub><mi>N</mi><mi>S</mi></msub><mo>=</mo><mn>0</mn></mrow></mfenced><mo>+</mo><mtext>P</mtext><mfenced><mrow><msub><mi>N</mi><mi>S</mi></msub><mo>≥</mo><mn>1</mn></mrow></mfenced><mtext>P</mtext><mfenced><mrow><msub><mi>N</mi><mi>R</mi></msub><mo>=</mo><mn>0</mn></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mtext>one of Rachel or Sophia qualify</mtext></mfenced><mo>=</mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>4772</mn><mo>…</mo><mo>×</mo><mn>0</mn><mo>.</mo><mn>6504</mn></mrow></mfenced><mo>+</mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>3495</mn><mo>…</mo><mo>×</mo><mn>0</mn><mo>.</mo><mn>5227</mn><mo>…</mo></mrow></mfenced></math></p>
<p><br><strong>OR</strong></p>
<p>uses <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><msub><mi>N</mi><mi>R</mi></msub><mo>≥</mo><mn>1</mn></mrow></mfenced><mo>+</mo><mtext>P</mtext><mfenced><mrow><msub><mi>N</mi><mi>S</mi></msub><mo>≥</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mn>2</mn><mo>×</mo><mtext>P</mtext><mfenced><mrow><msub><mi>N</mi><mi>R</mi></msub><mo>≥</mo><mn>1</mn></mrow></mfenced><mo>×</mo><mtext>P</mtext><mfenced><mrow><msub><mi>N</mi><mi>S</mi></msub><mo>≥</mo><mn>1</mn></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mtext>one of Rachel or Sophia qualify</mtext></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>4772</mn><mo>…</mo><mo>+</mo><mn>0</mn><mo>.</mo><mn>3495</mn><mo>…</mo><mo>-</mo><mn>2</mn><mo>×</mo><mn>0</mn><mo>.</mo><mn>4772</mn><mo>…</mo><mo>×</mo><mn>0</mn><mo>.</mo><mn>3495</mn><mo>…</mo></math></p>
<p> </p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>4931</mn><mo>…</mo></math></p>
<p> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>493</mn></math> <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> <em><strong>M</strong></em> marks are not dependent on the previous <em><strong>A</strong></em> marks.</p>
<p> </p>
<p><em><strong>[7 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p>In general, well answered with many candidates getting full marks. On the other end of the spectrum, some candidates could only get the first two marks for normal distribution without recognising the binomial distribution and often just stopping there. Some candidates lost the last two marks for not being able to combine the probabilities correctly.</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>
<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="data:image/png;base64,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"></p>
<p>normal curve centred on 50 <em><strong>A1</strong></em></p>
<p>vertical lines at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> = 42 and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> = 54, with shading in between <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>P(42<em> </em>< <em>X</em> < 54) (= P(− 2<em> </em>< <em>Z</em> < 1)) <em><strong>(M1)</strong></em></p>
<p>= 0.819 <em><strong> A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>P(<em>μ</em> − <em>k</em>σ < <em>X</em> < <em>μ</em> + <em>k</em>σ) = 0.5 ⇒ P(<em>X</em> < <em>μ</em> + <em>k</em>σ) = 0.75 <em><strong>(M1)</strong></em></p>
<p><em>k</em> = 0.674 <em><strong> A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1A0</strong></em> for <em>k</em> = −0.674<em>.</em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</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>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<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><em>X</em> ~ N(820, 230<sup>2</sup>) <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award<em><strong> M1</strong></em> for an attempt to use normal distribution. Accept labelled normal graph.</p>
<p>⇒P(<em>X</em> > 1000) = 0.217 <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>Y</em> ~ B(24,0.217...) <em><strong> (M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for recognition of binomial distribution with parameters.</p>
<p>P(<em>Y</em> ≤ 10) − P(<em>Y</em> ≤ 4) <em><strong> (M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for an attempt to find P(5 ≤ <em>Y</em> ≤ 10) or P(<em>Y</em> ≤ 10) − P(<em>Y</em> ≤ 4).</p>
<p>= 0.613 <em><strong>A1</strong></em></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="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>
<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 <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="0.5 = k + 3k - {k^2}">
<mn>0.5</mn>
<mo>=</mo>
<mi>k</mi>
<mo>+</mo>
<mn>3</mn>
<mi>k</mi>
<mo>−</mo>
<mrow>
<msup>
<mi>k</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{k^2} - 4k + 0.5 = 0">
<mrow>
<msup>
<mi>k</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>4</mn>
<mi>k</mi>
<mo>+</mo>
<mn>0.5</mn>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = 0.129">
<mi>k</mi>
<mo>=</mo>
<mn>0.129</mn>
</math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Do not award the final <strong><em>A1 </em></strong>if two solutions are given.</p>
<p> </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>use of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A' \cap B) = {\text{P}}(B) - {\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>
<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> or alternative <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A' \cap B) = 3k - {k^2}">
<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>
<mn>3</mn>
<mi>k</mi>
<mo>−</mo>
<mrow>
<msup>
<mi>k</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 0.371">
<mo>=</mo>
<mn>0.371</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[3 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 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>
<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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Χ</mi><mo>~</mo><mtext>N</mtext><mfenced><mrow><mn>102</mn><mo>,</mo><mo> </mo><msup><mn>8</mn><mn>2</mn></msup></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>Χ</mi><mo><</mo><mn>100</mn></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>401</mn></math> <em><strong>(M1)</strong></em><em><strong>A1</strong></em></p>
<p><em><strong><br>[2 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>P</mtext><mfenced><mrow><mi>Χ</mi><mo>></mo><mi>w</mi></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>444</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>w</mi><mo>=</mo><mn>103</mn><mo> </mo><mfenced><mtext>g</mtext></mfenced></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>Χ</mi><mo>></mo><mn>100</mn><mo> </mo><menclose notation="left"><mo> </mo><mi>Χ</mi><mo>></mo><mn>105</mn></menclose></mrow></mfenced><mo>=</mo><mfrac><mrow><mtext>P</mtext><mfenced><mrow><mi>Χ</mi><mo>></mo><mn>100</mn><mo>∩</mo><mi>Χ</mi><mo>></mo><mn>105</mn></mrow></mfenced></mrow><mrow><mtext>P</mtext><mfenced><mrow><mi>Χ</mi><mo>></mo><mn>105</mn></mrow></mfenced></mrow></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mtext>P</mtext><mfenced><mrow><mi>Χ</mi><mo>></mo><mn>100</mn></mrow></mfenced></mrow><mrow><mtext>P</mtext><mfenced><mrow><mi>Χ</mi><mo>></mo><mn>105</mn></mrow></mfenced></mrow></mfrac></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>0</mn><mo>.</mo><mn>15865</mn><mo>…</mo></mrow><mrow><mn>0</mn><mo>.</mo><mn>35383</mn><mo>…</mo></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>448</mn></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mn>90</mn><mo><</mo><mi>Χ</mi><mo><</mo><mn>114</mn></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>866</mn><mo>…</mo></math> <em><strong>(A1)</strong></em></p>
<p><br><strong>OR</strong></p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mo>-</mo><mn>1</mn><mo>.</mo><mn>5</mn><mo><</mo><mi>Z</mi><mo><</mo><mn>1</mn><mo>.</mo><mn>5</mn></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>866</mn><mo>…</mo></math> <em><strong>(A1)</strong></em></p>
<p><br><strong>THEN</strong></p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>866</mn><mo>…</mo><mo>×</mo><mn>500</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>433</mn></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[3 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mtext>P</mtext><mfenced><mrow><mi>Χ</mi><mo><</mo><mn>95</mn></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>19078</mn><mo>…</mo></math> <em><strong>(A1)</strong></em></p>
<p>recognising <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo>~</mo><mtext>B</mtext><mfenced><mrow><mn>80</mn><mo>,</mo><mo> </mo><mi>p</mi></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p>now using <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo>~</mo><mtext>B</mtext><mfenced><mrow><mn>80</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>.</mo><mn>19078</mn><mo>…</mo></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>Y</mi><mo>≥</mo><mn>20</mn></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>116</mn></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[4 marks]</strong></em></p>
<div class="question_part_label">e.</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>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<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>let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span> be the random variable “amount of caffeine content in coffee”</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(X > 120) = 0.2,{\text{ P}}(X > 110) = 0.6">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>></mo>
<mn>120</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0.2</mn>
<mo>,</mo>
<mrow>
<mtext> P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>></mo>
<mn>110</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0.6</mn>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="( \Rightarrow {\text{P}}(X < 120) = 0.8,{\text{ P}}(X < 110) = 0.4)">
<mo stretchy="false">(</mo>
<mo stretchy="false">⇒</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo><</mo>
<mn>120</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0.8</mn>
<mo>,</mo>
<mrow>
<mtext> P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo><</mo>
<mn>110</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0.4</mn>
<mo stretchy="false">)</mo>
</math></span></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>M1 </em></strong>for at least one correct probability statement.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{120 - \mu }}{\sigma } = 0.84162 \ldots ,{\text{ }}\frac{{110 - \mu }}{\sigma } = - 0.253347 \ldots ">
<mfrac>
<mrow>
<mn>120</mn>
<mo>−</mo>
<mi>μ</mi>
</mrow>
<mi>σ</mi>
</mfrac>
<mo>=</mo>
<mn>0.84162</mn>
<mo>…</mo>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mfrac>
<mrow>
<mn>110</mn>
<mo>−</mo>
<mi>μ</mi>
</mrow>
<mi>σ</mi>
</mfrac>
<mo>=</mo>
<mo>−</mo>
<mn>0.253347</mn>
<mo>…</mo>
</math></span> <strong><em>(M1)(A1)(A1)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>M1 </em></strong>for attempt to find at least one appropriate <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z">
<mi>z</mi>
</math></span>-value.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="120 - \mu = 0.84162\sigma ,{\text{ }}110 - \mu = - 0.253347\sigma ">
<mn>120</mn>
<mo>−</mo>
<mi>μ</mi>
<mo>=</mo>
<mn>0.84162</mn>
<mi>σ</mi>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>110</mn>
<mo>−</mo>
<mi>μ</mi>
<mo>=</mo>
<mo>−</mo>
<mn>0.253347</mn>
<mi>σ</mi>
</math></span></p>
<p>attempt to solve simultaneous equations <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu = 112,{\text{ }}\sigma = 9.13">
<mi>μ</mi>
<mo>=</mo>
<mn>112</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>σ</mi>
<mo>=</mo>
<mn>9.13</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[6 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</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>
<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="{\text{E}}({X^2}) = \Sigma {x^2} \bullet {\text{P}}(X = x) = 10.37{\text{ }}( = 10.4{\text{ 3 sf)}}">
<mrow>
<mtext>E</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>X</mi>
<mn>2</mn>
</msup>
</mrow>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi mathvariant="normal">Σ</mi>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<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>
<mn>10.37</mn>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mo>=</mo>
<mn>10.4</mn>
<mrow>
<mtext> 3 sf)</mtext>
</mrow>
</math></span> <strong><em>(M1)A1</em></strong></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><strong>METHOD 1</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sd}}(X) = 1.44069 \ldots ">
<mrow>
<mtext>sd</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>1.44069</mn>
<mo>…</mo>
</math></span> <strong><em>(M1)(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{Var}}(X) = 2.08{\text{ }}( = 2.0756)">
<mrow>
<mtext>Var</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>2.08</mn>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mo>=</mo>
<mn>2.0756</mn>
<mo stretchy="false">)</mo>
</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{E}}(X) = 2.88{\text{ }}( = 0.06 + 0.27 + 0.5 + 0.98 + 0.63 + 0.44)">
<mrow>
<mtext>E</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>2.88</mn>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mo>=</mo>
<mn>0.06</mn>
<mo>+</mo>
<mn>0.27</mn>
<mo>+</mo>
<mn>0.5</mn>
<mo>+</mo>
<mn>0.98</mn>
<mo>+</mo>
<mn>0.63</mn>
<mo>+</mo>
<mn>0.44</mn>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(A1)</em></strong></p>
<p>use of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{Var}}(X) = {\text{E}}({X^2}) - {\left( {{\text{E}}(X)} \right)^2}">
<mrow>
<mtext>Var</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<mtext>E</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>X</mi>
<mn>2</mn>
</msup>
</mrow>
<mo stretchy="false">)</mo>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>E</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span> <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(M1) </em></strong>only if <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {{\text{E}}(X)} \right)^2}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>E</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span> is used correctly.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {{\text{Var}}(X) = 10.37 - 8.29} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>Var</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>10.37</mn>
<mo>−</mo>
<mn>8.29</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{Var}}(X) = 2.08{\text{ }}( = 2.0756)">
<mrow>
<mtext>Var</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>2.08</mn>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mo>=</mo>
<mn>2.0756</mn>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Accept 2.11.</p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{E}}(X) = 2.88{\text{ }}( = 0.06 + 0.27 + 0.5 + 0.98 + 0.63 + 0.44)">
<mrow>
<mtext>E</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>2.88</mn>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mo>=</mo>
<mn>0.06</mn>
<mo>+</mo>
<mn>0.27</mn>
<mo>+</mo>
<mn>0.5</mn>
<mo>+</mo>
<mn>0.98</mn>
<mo>+</mo>
<mn>0.63</mn>
<mo>+</mo>
<mn>0.44</mn>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(A1)</em></strong></p>
<p>use of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{Var}}(X) = {\text{E}}\left( {{{\left( {X - {\text{E}}(X)} \right)}^2}} \right)">
<mrow>
<mtext>Var</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<mtext>E</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo>−</mo>
<mrow>
<mtext>E</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</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="(0.679728 + \ldots + 0.549152)">
<mo stretchy="false">(</mo>
<mn>0.679728</mn>
<mo>+</mo>
<mo>…</mo>
<mo>+</mo>
<mn>0.549152</mn>
<mo stretchy="false">)</mo>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{Var}}(E) = 2.08{\text{ }}( = 2.0756)">
<mrow>
<mtext>Var</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>E</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>2.08</mn>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mo>=</mo>
<mn>2.0756</mn>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[3 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 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>
<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="X \sim {\text{Po}}\left( {5.3} \right)">
<mi>X</mi>
<mo>∼</mo>
<mrow>
<mtext>Po</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>5.3</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {X = 4} \right) = {{\text{e}}^{ - 5.3}}\frac{{{{5.3}^4}}}{{4{\text{!}}}}">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo>=</mo>
<mn>4</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>5.3</mn>
</mrow>
</msup>
</mrow>
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mn>5.3</mn>
</mrow>
<mn>4</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>4</mn>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
</mfrac>
</math></span> <em><strong>(M1)</strong></em></p>
<p>= 0.164 <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>listing probabilities (table or graph) <em><strong>M1</strong></em></p>
<p>mode <em>X</em> = 5 (with probability 0.174) <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M0A0</strong></em> for 5 (taxis) or mode = 5 with no justification.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>mode is the integer part of mean <em><strong>R1</strong></em></p>
<p>E(<em>X</em>) = 5.3 ⇒ mode = 5 <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Do not allow <em><strong>R0A1</strong></em>.</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt at conditional probability <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{P}}\left( {X = 7} \right)}}{{{\text{P}}\left( {X \geqslant 6} \right)}}">
<mfrac>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo>=</mo>
<mn>7</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo>⩾</mo>
<mn>6</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
</math></span> or equivalent <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( { = \frac{{0.1163 \ldots }}{{0.4365 \ldots }}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>0.1163</mn>
<mo>…</mo>
</mrow>
<mrow>
<mn>0.4365</mn>
<mo>…</mo>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p>= 0.267 <em><strong> A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><em><strong>METHOD 1</strong></em></p>
<p>the possible arrivals are (2,0), (1,1), (0,2) <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="Y \sim {\text{Po}}\left( {0.65} \right)">
<mi>Y</mi>
<mo>∼</mo>
<mrow>
<mtext>Po</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>0.65</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p>attempt to compute, using sum and product rule, <em><strong>(M1)</strong></em></p>
<p>0.070106… × 0.52204… + 0.026455… × 0.33932… + 0.0049916… × 0.11028… <em><strong>(A1)(A1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong> </em>for one correct product and <em><strong>A1</strong> </em>for two other correct products.</p>
<p>= 0.0461 <em><strong>A1</strong></em></p>
<p><em><strong>[6 marks]</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>recognising a sum of 2 independent Poisson variables <em>eg Z</em> = <em>X</em> + <em>Y</em> <em><strong>R1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda = 5.3 + \frac{{1.3}}{2}">
<mi>λ</mi>
<mo>=</mo>
<mn>5.3</mn>
<mo>+</mo>
<mfrac>
<mrow>
<mn>1.3</mn>
</mrow>
<mn>2</mn>
</mfrac>
</math></span></p>
<p>P(<em>Z</em> = 2) = 0.0461 <em><strong>(M1)A3</strong></em></p>
<p><em><strong>[6 marks]</strong></em></p>
<p> </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.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.iii.</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>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>
<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="{\text{P}}(5{\text{ or more}}) = \frac{{29}}{{75}}\,\,\,( = 0.387)"> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mn>5</mn> <mrow> <mtext> or more</mtext> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mfrac> <mrow> <mn>29</mn> </mrow> <mrow> <mn>75</mn> </mrow> </mfrac> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mo stretchy="false">(</mo> <mo>=</mo> <mn>0.387</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">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{mean score}} = \frac{{2 \times 3 + 3 \times 15 + 4 \times 28 + 5 \times 17 + 6 \times 9 + 7 \times 3}}{{75}}"> <mrow> <mtext>mean score</mtext> </mrow> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mo>×</mo> <mn>3</mn> <mo>+</mo> <mn>3</mn> <mo>×</mo> <mn>15</mn> <mo>+</mo> <mn>4</mn> <mo>×</mo> <mn>28</mn> <mo>+</mo> <mn>5</mn> <mo>×</mo> <mn>17</mn> <mo>+</mo> <mn>6</mn> <mo>×</mo> <mn>9</mn> <mo>+</mo> <mn>7</mn> <mo>×</mo> <mn>3</mn> </mrow> <mrow> <mn>75</mn> </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{{323}}{{75}}\,\,\,( = 4.31)"> <mo>=</mo> <mfrac> <mrow> <mn>323</mn> </mrow> <mrow> <mn>75</mn> </mrow> </mfrac> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mo stretchy="false">(</mo> <mo>=</mo> <mn>4.31</mn> <mo stretchy="false">)</mo> </math></span> <strong><em>A1</em></strong></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 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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>6.75 <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>2.22 <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>8.75 <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>2.22 <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>the order is 3, 4, 6, 7, 7, 8, 9, 10</p>
<p>median is currently 7 <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> This can be indicated by a diagram/list, rather than actually stated.</p>
<p>with 9 numbers the middle value (median) will be the 5th value <em><strong> R1</strong></em></p>
<p>which will correspond to 7 regardless of whether the position of the median moves up or down <em><strong> R1</strong></em></p>
<p><strong>Note:</strong> Accept answers using data 5, 6, 8, 9, 9, 10, 11, 12 (ie from part (b)).</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</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.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</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.</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>
<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="{\text{P}}(A|B) = \frac{{{\text{P}}(A \cap B)}}{{{\text{P}}(B)}}">
<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>
<mfrac>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo>∩</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow 0.75 = \frac{{0.6}}{{{\text{P}}(B)}}">
<mo stretchy="false">⇒</mo>
<mn>0.75</mn>
<mo>=</mo>
<mfrac>
<mrow>
<mn>0.6</mn>
</mrow>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow {\text{P}}(B){\text{ }}\left( { = \frac{{0.6}}{{0.75}}} \right) = 0.8">
<mo stretchy="false">⇒</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>0.6</mn>
</mrow>
<mrow>
<mn>0.75</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0.8</mn>
</math></span> <strong><em>A1</em></strong></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{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></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow 0.95 = {\text{P}}(A) + 0.8 - 0.6">
<mo stretchy="false">⇒</mo>
<mn>0.95</mn>
<mo>=</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mn>0.8</mn>
<mo>−</mo>
<mn>0.6</mn>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow {\text{P}}(A) = 0.75">
<mo stretchy="false">⇒</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0.75</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</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}}(A'|B) = \frac{{{\text{P}}(A' \cap B)}}{{{\text{P}}(B)}} = \frac{{0.2}}{{0.8}} = 0.25">
<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>
<mfrac>
<mrow>
<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>
</mrow>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mn>0.2</mn>
</mrow>
<mrow>
<mn>0.8</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0.25</mn>
</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'|B) = {\text{P}}(A’)">
<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>
<msup>
<mi>A</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">)</mo>
</math></span> <strong><em>R1</em></strong></p>
<p>hence <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 <strong><em>AG</em></strong></p>
<p> </p>
<p><strong>Note:</strong> If there is evidence that the student has calculated <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A' \cap B) = 0.2">
<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>
<mn>0.2</mn>
</math></span> by assuming independence in the first place, award <strong><em>A0R0</em></strong>.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A) = {\text{P}}(A|B)">
<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>
<mrow>
<mo stretchy="false">|</mo>
</mrow>
<mi>B</mi>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A) \times {\text{P}}(B) = 0.75 \times 0.80 = 0.6 = {\text{P}}(A \cap B)">
<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>
<mn>0.75</mn>
<mo>×</mo>
<mn>0.80</mn>
<mo>=</mo>
<mn>0.6</mn>
<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>A1</em></strong></p>
<p><strong>THEN</strong></p>
<p><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 independent <strong><em>R1</em></strong></p>
<p>hence <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 <strong><em>AG</em></strong></p>
<p><strong>METHOD 3</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A') \times {\text{P}}(B) = 0.25 \times 0.80 = 0.2">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<msup>
<mi>A</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">)</mo>
<mo>×</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0.25</mn>
<mo>×</mo>
<mn>0.80</mn>
<mo>=</mo>
<mn>0.2</mn>
</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') \times {\text{P}}(B) = {\text{P}}(A' \cap B)">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<msup>
<mi>A</mi>
<mo>′</mo>
</msup>
<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>
<msup>
<mi>A</mi>
<mo>′</mo>
</msup>
<mo>∩</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
</math></span> <strong><em>R1</em></strong></p>
<p>hence <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 <strong><em>AG</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">c.</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>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<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 inverse normal (implied by ±0.1509… or ±1.554…) <em><strong>(M1)</strong></em></p>
<p>P(<em>X</em> < 16) = 0.56</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \frac{{16 - \mu }}{\sigma } = 0.1509 \ldots ">
<mo stretchy="false">⇒</mo>
<mfrac>
<mrow>
<mn>16</mn>
<mo>−</mo>
<mi>μ</mi>
</mrow>
<mi>σ</mi>
</mfrac>
<mo>=</mo>
<mn>0.1509</mn>
<mo>…</mo>
</math></span> <em><strong>(A1)</strong></em></p>
<p>P(<em>X</em> < 17) = 0.94</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \frac{{17 - \mu }}{\sigma } = 1.554 \ldots ">
<mo stretchy="false">⇒</mo>
<mfrac>
<mrow>
<mn>17</mn>
<mo>−</mo>
<mi>μ</mi>
</mrow>
<mi>σ</mi>
</mfrac>
<mo>=</mo>
<mn>1.554</mn>
<mo>…</mo>
</math></span> <em><strong>(A1)</strong></em></p>
<p>attempt to solve a pair of simultaneous equations <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu ">
<mi>μ</mi>
</math></span> = 15.9, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sigma ">
<mi>σ</mi>
</math></span> = 0.712 <em><strong>A1A1</strong></em></p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correctly shaded diagram or intent to find P(<em>X</em> ≥ 15) <em><strong> (M1)</strong></em></p>
<p>= 0.895 <em><strong> A1</strong></em></p>
<p><strong>Note:</strong> Accept answers rounding to 0.89 or 0.90. Award <em><strong>M1A0</strong></em> for the answer 0.9.</p>
<p> </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><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,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"></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>
<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="X \sim {\text{Po}}\left( {2.1} \right)">
<mi>X</mi>
<mo>∼</mo>
<mrow>
<mtext>Po</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>2.1</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {X = 0} \right) = 0.122\left( { = {{\text{e}}^{ - 2.1}}} \right)">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0.122</mn>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>2.1</mn>
</mrow>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong> (M1)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><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAkUAAAB2CAYAAADGD+frAAAgAElEQVR4Ae2dDVwU17n/f7taayyi15rEXc2VCAFtJDGBi/8oaRaMEGsSDSQm1fBy4W/jrSZpuALF1NRWo1W4WG1Mk3qhIM172WqMsaCSTWLSQtnUBFKFgsUEdkm1VmFrlMjM/cyyC7Ozs7AvM7MvPHw+fHbmzJnnPOf7nJnznJc5R8WyLAv6IwJEgAgQASJABIjAKCegHuX5p+wTASJABIgAESACRMBKgJwiKghEgAgQASJABIgAEQBAThEVAyJABIgAESACRIAICJ0ilUpFUIgAESACRIAIEAEiMCoICP2escJcCyMIr9M5ESACRIAIEAEiQARCkYCTU0Qfo7lnZs55JFbuseJiES/3WckZk+wgJ13/yibb+pe/nKmTbeWjy7Hl/9GcIj4NOiYCRIAIEAEiQARGLQFyikat6SnjRIAIEAEiQASIAJ8AOUV8GnRMBIgAESACRIAIjFoC0jlFllbU6feiMPUH0Hf12YAysBh3Ikm7jhc2allTxokAESACRIAIEIEAJiCNU9RTh8LoGCxK/x521L6OF2tOg7FmWo0J02ZiFv6I+paeAMYgp2oMLK01KMmKtU42ViVtgL51tLJwjzNjNuIA52AnxSJL/7l7N1EsSQkw5gZUFqYOlFmVCtqsnThC5VZSxv4T1oNW/QYkqVRW+2qz9qDBbG/I+k8rSllqAhdgLLkXqqgSGK9KLTt05UnjFIUnY7uJRX9nNXI0jrDU05dj58v5WBAT7nhhlJwxXfvxpG49mu56Bb3sRZzM+ALrdBuo58yV/a8aUXpnPJZzDrbhS1exKFxOApYGlP5wP7CiAv1sP3pbqpHxWTFSqNzKSV0h2X3o0v8CFcjEwUHbPov52b9B60BLViE9KBl5CXCjNL/G+vxD8iYTgtKlcYpsYNSab2FhLNDUYoLFDotpw9EzUVg0fZw9ZBT9noNh91aUxz6Fp7PnIgzhmJ2djy2xeqzbfRzUXyRSFMbGYX0bi68aixEpcpmC5CbAoKfhTxjzRAGy4jRQQ42w6OV4emM2NGY9rxdYbj1IviwEej7EK2cWY0PabIRZbfsd5GYkArUNaP6CuhNkYe4HodbG+MO/BRYIein8oEuwJSmpU2TPvPlEB7qtrY4+dB08jjF3xyLMfnE0/fZ8gpoqIyIX34pZdtLqCCQ+nAhz1VE09lDTbDQVh+DIqxrhyWvxVNxknrpqhM+Jx2JeCB0GKYFwHdY/lcB7H1vQ2XIampwUzL/eadm6IM3kKFebOYP9z/wC2PEs1kZNGOUwPM++var2/E6xO9RTETFPO3iF6TqKaizCfaOylwi4+tePUG2OxMJZ12HodTMWEydPAcx/wEd/vTTIig6IQCATYHovoBu3Y/Gt02jDxEA2lAe6cXP39CX/je91P466XcsxXdrawANNKKp0BPrQtb8YBXgcP10eSc+qF2CH6movbnZ5S1MbOnvOoLNuIh5aNXOUGuYqzp5pQzsmI33yNSKoenH24mWA12YTiURBRCAACFxG2/HfoyknB+W38XuQAkA1UsELAldh1q+DNv3FgXs1wOG/pCA6gRsupb9gJsB0vYVnCoAd796L6eovgjkrftNd4mcgDDNiZgHoQsMbJ/BvDyyERuIU/EbK64Qn4tpJ472+m24kAv4mwHS9je1br8NzP+VetP7WhtL3ncBYaNJeAMteREttKTKxD3nzV6PUeMF30STBfwSsw2bVmPPaFqSN0tEZKeDL9IqbiOjvLMFtYTKJlyLnsssYi2tnRiESn6PpzD9FUiNnSQQKBQUaAeuL9lVc/6siLKcXbaBZx0d9whG9+En84jdF0OAjHPm427aUio9i6XY/EOCGzXbixTlPYo3DfEA/qBLkSUrstYzDtFn3oaJuK3mqAMbedDvSNZfQfeFfvJfNZZhOtwCaO3D7TTQJLsifnxBX/wKMpcVoXLUTW5Kn09BKSFpbjfD4u5FBHykFuXW/QP2Bt1CbPx8TbetPqVT/jvR97UB7PuK/pkJUiRH0feHIZpbQKeIWKfwAjdplyJo9OtckcsIdfgtSM7Sofe1DtNk/NGM68fGRFmgy7kZ8uIT4nRKnACLgC4EenCrfhOenrMEGu0PEdKFuw3+jvJWbC0d/oUKA6e7ACTNNog9ue96AtMo2sCzL+/8M1ZmRQGQxGr9i0bY+jvfBT3DnVk7tfayVuRUzH0Bq+cfoMv4ex3qjoIsmh2jIYFOhe2IDcpp24tmKZljQg1MVxfhRUxqeeyIRRGqIlOMRg0sXz+MShL1sjrHoTCYCjBkNOx9Hcu4ulOfGDrU8x8zAotc1uHUWzZGTibzMYm3bLqlikVVSg1YLA1iaUfHs8+grehKraBK9zPxJfDAQ8M0pYi7izMk2NL1bh79MTMQy62JvwZBt5XTkVvTeZfgJplWlYKJqEpLfnYtKAw0vurbA59BnRWPSom0ww4za3DkYQ8vUu8Yl+RVuyGw15uftg1lEtib9dtwkzzerIqlRkLQE1JignYs7dP/Avvx7EDNxDFT3VQFZL+Pg1sX0UYy0sElakBJQsVx/m+1PpVJZu97s5/TrmgCxcs1G7ArxEqOifBjZQXnmSqVItlWKtPLpkG3lYy5k61tPkXx6kmQiQASIABEgAkSACChKgJwiRXFTYkSACBABIkAEiECgEiCnKFAtQ3oRASJABIgAESACihIgp0hR3JQYESACRIAIEAEiEKgEnCZaB6qipBcRIAJEgAgQASJABKQmwPvezHktJ/5FqRMOJXnCGeuhlDc58kK85KDquUyyg+fMguUOsm2wWMpzPcm2njNz9w6OLf+Phs/4NOiYCBABIkAEiAARGLUEyCkataanjBMBIkAEiAARIAJ8AuQU8WnQMREgAkSACBABIjBqCXjtFF01liBqcDdeFbhxOet/UiHK61phEUNqacWRkjXI1X8O4CrM+jVD96lsu/j21KFQa5eXhBKjqCQx6dKEmfXIzdqJI6090sgTSGHMDagsTB1kVWk0w75XrCCq4LQHrXUHoC8vRFJqOVrFbmLMMFYWIolnCzH5DjqotEgqrILR3CdILzBOHXRNKoRYfpw17YPZWIXCJK2tfKWisLIBZjFm4Pbvu9ehHKq0G1DXY4vMnEJ5ql2OvVzaf+NRWHfOOfmQC+HzdL+8ONhOrJx5wNZBltvlIOQMIUuGvGPLbQBugF6/F4VJq8Q3Cba+77Og5b2PnOsG7r1WPvSskm1lsTEJ9YAAt82H/Q/gdvnw4O/iMbZAAzayuJH9imXZftNxtjRzLgvMZXOqO9h+vqjeJrYsU8dmlh5nTYMX+tnexlJWh0g2s/qzwdj9ndXsYznPsfWmK4Nhyh1cZFuqi1idJoctO3nRZbIes+Ik9dazxbqlbEH1SbbXyquWLdItZYuOdTqyckr1S7alLI/NK1jJagAWKWVsyyBDe+R/so3FS7ktWwT/AltYddAI4oCFZi1b3Skfb+V42cuUkANYTU412yng1t9SxqYImGkKjrF2y4tdH2Qsage7PQLz13M72HjqitjqFo7KFdZ0bBOr021ijw33fLpRztxm6/VzE5g2kEsrz23r/Tupv6WCzcnLYzM13HP2EFvW8qVjtvo72OqcOFZXUM229HIPXT/b21LNFujieHXDRfZkWQ4bmWl/119hTfXPsZkawTvLUfKoPPPKtv0mtrGigNVZ328ani2GQdjbwh6rfokt5urxzGrW5DIqZ8932DeKMwfqJOjY4kauVuPKlLsyXApX9IKQrYMXJLw4omZfNbLFkUNOERefc2hyuAeFX2HYH5Dieqsz4CC3/yRblqJhByui3nq2tGAfe9L6IDnEVPDE/uIvZRtd6OExK3bAaRnMpzU3/ezFY0WsRuc6HYdM21g5sLVF4CqYpYMvF6vHNfRADNqCc65yHBzTflM9W1GQwgIaNqXs5AjOmYM2Hp0oxotjtDSHLa032fJyhTU17mMLdJwjKHx5n2WPFWSyxY3/dJGXL9mWii08WfZonN02sktl5GVPSepfj+1gdUgWsAXHzvJU4bgtYHViz7M1ljvlzF22Ejw3PM1D+dBj2/r8TuLs/JDIc2V7rzk9b7b4tvfRQF0hfCavsJ3Va1mNpog9dlHQggll442QN49t29/JHitKYWFvvFg7JeaymsyKYerWz9jqTB2r00UONJpdOUX9Jra+lHOGOEfrV+z+Rvu7lsuEmzJGyK+Sl4VsJXeKWJujhMhitpHrPuJaCFxvkMtCzr1g41hYr/+DbSzdJmuPhduwh3PkBtYtcFuUNaK1V03E8bCGxwkqHReiXTpFX7It1W+IOHB8tv0s29/CVlc0Ojumth4/R4fNhQ5eBgsL3ohivOTV33KQrXBycuwvaUfOgz0VugK2rPodW4uWr9lFtq3lCxFHkeP6mHPrmH9rgB57Zgc7N2HFZQt39Uy7Vc7cZOtlOQhQ/LKq5ZltWZb1ma0rp4h71XA9sI7PG8ty8VfanGnX97rUS1Z6gS3cM9vae8sd+Q/YRKQOEmbdVM1mcr1LYk6R3dlCyvAjHMPJEKbn53MhW6/nFLkcobt0EWcvAYicgolW6efR8PpLMCyOx5xwseSmID41BRrzQZQ9/hT2zvwulk8f51K8bxf485juRYnxwoA46zymFY7j4urrMXfhLBhK96PBPr/E68QZ9DQeRZVZi3kRU+FAIUyLmFgTqmo+gfezmMYjOu1BxIU5SB7UVpNxN+I59upopGXFIWzwCv8gDhmptyCcH+S3Y+95qaPvRVbcZHHNNSlIjZ9iu3YOhrLnUcudGXYgNz0JMdH/iZ0N/Dle4YiMvs7RXlz8nk9Q82kCEqPGi6cTMqHn0VhTC7MmChHT+M+kGmEzohBrrkVN43nn3LpVztxh6305cFaKQhwJyMtWHbUIj+VcwY5Fq1CoPwULGFhOvYbtH6Tgl2viXbyD+Bqa0dRiEp+byo9GxyIEbHUu4jF/ztC7UB21AA+nALWvfYg20fmVIqIcgvrQtX8bHt1mQk71r7Alebrzu9EhfnCeiNeiHublUlMz2iwMYDkF/bPbscM8FzmPLUIUJ/3qGXxUbURk7ExcKypXjfA58ViMf8AUuxYlaTNHAP059FlRjhNj7RP5bL9RJUZcFU1rLDRpL4C9eAwFmo9w5OPugUnOYTFIzZ6I8738u8Zj1q0JiDS3oaPb10nIfejuaIMZsxAzQ9wlMZ/oQLdXBVU0owOBXOVdNQtbchcM4+zYXo6x30eubuowwpS8JDUvrnJ/F7FbMqEbdMynInl7I1j2IlqOVaOsIAUw70Pe/NUotTvLolnmeBnwadqCgfItGidEAplz6DhhAmKjMEPU4TbhRMc5Nz8UcKecCdlKXQ5CxC6SZENmtuqZSNv1CsoyTdiRnoz7sorwQv0cbNmbhdnWsjQO0yKioMFJfND8hZtlSJKMh74QW52LyCjMvHbsUH7V38DkaROA2gY0f8Gv64aiDHvUcxy71+2BWXM75vzjeSyy1bfarD1oCNAPdYbNj4uLkjhF5n3ZmDNxDFQT5yC9ahqKa6uxy+7cWHuOIrFw1nXOy2dblWLQc7IRR5CIjPtvcaMFcQPSKtu4YT+X/23r41ykZaMwYRKunWBG+/l/2R7Gq7iA/0BSzAQRTKfR0qnwF3AiWnge1Ieuo2/iRN56rIgepkeD+RxHXzqDvK1piJakNHiuqdx3MF3v4aUTD2LrimgRhzsc0clpyNl+EKZjm6DDIZQ+X4culw4q52D9HWmJESKy5M5JEMt3q5wR2yC2sLPqYXORvfUnyNGYYdi3A/kV+/GHNnt/uBrhCcuRp/sHytc9g132HlruC9rf1aDBrEFsjNaN+sA5WQqxEVg4C1qeTwT8G2bG3uAlHlujxqyBLuMu3H7X03iHa1BWFyFm3zrMX7kHRq5jJAT+JKkGI4sb8ZXdSTFVYv3iaEFhvoTuC3YHREjNVRe9MJ6E52Ovw6yFkbh09iIugUGP4S2cSX/E5fCThCkrJorpegubD92GXcN2VXPdoc/j0MIirHE15KSYxjIlxJzB/s3vYeGu3BHsOw6a5EK8WPYQzIcb0eLqAR81Q2dS2sPNckZspYTuZ1nccFklVm8Gftp5Bab655DZsg3puqdQfsrmGIUlIO/l/ajI6EbefC3GqFJRWNGM/olACxLxMDU8JLbhZVw82+ulTHvPohYJqfcjOZqbaBGO6LQ8bCyIAwwv4fUGkaF0L1Pz522SOEXDZsCpV0YQ29bVNzjvRXDZ+dSX4TO7tGswedpkmLsvoNdsQFnH/8Nql06B6yEvu7SRf+1dxSIxLSa0NJmhmReBaVJZw9KMit2nsWLLd21d1SLp2sb4d5+4G1uy5wqcWLH4SoZJxasHpyrKcGJFPrJnuzNbajyiEu9BisusCod3XEYMjQvqqYiYpxXJCwNLZxuaIDJHzin2wFySkcuZGFupyoGTUhQAmdn2GLA5+ReYuuLbmK4eB03Cf2GPoRoFMYeR++RvB9dZU2sSkLW9xtbrX4Pt2dHoPPQ2zCn3jII5ezIXQ65+c+i8+RIXum3zaD1O+ip6z58FMBHXTuKPPNjmBMOI6o/OuJi24nFifr3Bt2rYNql6oMfFRT7GzsTt6XFoP/IxTjsYiIvfh643f43S9kgsnn/TMPNe+LIlGD7DeEy6diLwWR12/7wV315xm4hTcBVnz7Sh3WmSKV8Xd4/VCI+/Gxka56E4prsDJ8wSTnJmulC39zgifvAEkjX8ybGOujJmA/Yeno4fbFoMjW+lwFGwJGdS8OqDue5lHI54DJs8nRDocg7NaBvesb3wmtrQ6dBzZms1OkxcFze8++VMjK0U5UBcLwqVk+1ltP72Beww8xuUaoRFL8fTG7Ohqf09jrddFjEB16NYjHXlM1EcwsP5IhmXNshW50L43DL/woXuS0BKAuZe7zCu5kb646GdFQOgF2cv8m2nxoRJUzABw02RcUN8AEXxujq0rmg9aRF2mAHzjkWYFFUCo+jcrSlIWLEKuqaP0MyfjGX94uvrmJG+B2a0Y1/6v8P1BGmpiY3FxCnXcn20mP9EjothlQs4Wd8EXd5yJAxOzvVBj/BbkJoBwVdmtha3bhVWJNi/ivIhDc4h2vQiOpasdHCIGHMddutbByczMuYj2PTzLixZreM5RJwTUQl9K7/A+6CLr7f6xIvLy278vCMRq/kOEcdn94HBVqqzin0wN7chrvB+8flVo254x1ZxQviVmQWdLV0jPhselTNXbH0qB84WphAeAdnY2nuhLuF8L/99YvtqUbShyT2z2/Bo+rtYXPaz0B3O5+GX79D+Rbfjc8u0fYjXar859BGURwqMxfXzU5CjMaLqpfd4cy4v4/THDWjH7bhrrvinVB4lEwiR+UsECL/X51/z6di6tsFSVldUy1vN2ieJPt7MLRy3bZgVq0daW4m1Lm7lqRL9Jv4K1vYVXgUrWlsXy4sUXxjP5TpF1lUzBxbrEqzOzNkUvPVCBnQQWdGaize47ox9nYulvIUNbStmOyw0KRYmTsWbsuUdL9vCm6IcMLhIKCd7Y8Eutta6SjOnM7eS+Ra2oKzJeR0na5a4tXlcLdgoPS9xir6Hem4H+0Km9hWsbSu+2xeFs6rkXA7cK2f2/AzHllspf6Tnxjn9gdXj+c+RSBx78iHy67ltvWQ7yGuYtYbsK5rzy4l9VwOHZ4y/MnLK4Gr/g0nYFpgE/73j9I4k2w7xsh1Z19mbO7h4o323iaHFG23MwH/H2+51ucaQbWFNzGUzrTa012H852w4Gfb3pGaofhssJ/YFjIfRyymT0gQInxuve4o8cujU05G84WfI6CrCyiI9Wh264j2S5H1k5gz0uako1L+PIzufw+/nZLmYZ9IHc8MvsfbhZmTU/RDJUvQS2bRWaxZjy8tFuLbyHoxRjcHEx+rxrZK9bqz3wKCnbgO0Y+Ygt9YM1OYiZowWqeWnbL0/51BXtAyLtllX3XFmZB/m6KlDUVwKthnMznEAuD+vS/R2yQM958Vx2oS4RZtgENVmaJhSPVGDiO69SImZBJV1X67XcHJmDrbluJpfJTa8I5pIiAVyE9CL8PL2qaiM+zpUqtl4rH4OSl4ucuiNdMi0x+VseLaelwMHbehkGAJes7X29F+DmNw3ALyB3JhroOLvyWidRH0Q1UtP41EtV25UUN1XBWS9iD22Z4xpLUcq9x6M2YY/TXkIB00HsT1ttshUhmEyQJfECViXRKhGrY3/GO0anIgtgWFPxjDzTC0wliRBpU3HPk7qvnRoVfz9R8dhetpWGGpzgB/FYiJnO90BXLvhbRxcn2Cz20gyxNUNpFAV52vZFeIKLu/UHizdL7dB4AuleHXW0yhL8/bTQC/V4TafXJKMH017Gvt/thoJrubbcBvC/vAMHnk6F4utM+zF05OdlXiyQRtKvALDdGSHwLCDHFqQbeWgGhgyybby2UHIVlmnSL58KS5ZCFJxBYIsQeIVGAYjOwSGHeTQgmwrB9XAkEm2lc8OQrbKDJ/Jlx+STASIABEgAkSACBABSQiQUyQJRhJCBIgAESACRIAIBDsBcoqC3YKkPxEgAkSACBABIiAJAXKKJMFIQogAESACRIAIEIFgJ0BOUbBbkPQnAkSACBABIkAEJCFATpEkGEkIESACRIAIEAEiEOwEyCkKdguS/kSACBABIkAEiIAkBMgpkgQjCSECRIAIEAEiQASCnQA5RcFuQdKfCBABIkAEiAARkIQAOUWSYCQhRIAIEAEiQASIQLATIKco2C1I+hMBIkAEiAARIAKSECCnSBKMJIQIEAEiQASIABEIdgLkFAW7BUl/IqA4gT6YjW9Df8AIM6N44kGeYKCzC3T9Atf8jNmIA/q3YTT3Ba6SpNmIBMgpGhGRHBH6YK6rhL71shzClZPJ/A11VX9UoGIMEV4yWoZpr0NVgxmy+yiMGQ2796D2G4lIWxYHDb1BPLTqOGjivoO0RV9HbWk5GgKpAiXbemhLx+hqTRyWpSXiG7V7sFuJZ9ExeTqTiAC90iQC6b4YBhbja3gVd2N59Hj3bwvEmOoboUs+j1ermmGRTb8Q4iUbI0AdmYhk8wFUneqRLxWmC3WbfonmlFxkzQ6XLx23JPfB3LAHWVoVVKpYZJWLlEGzHlkqFVRRJTBedUuob5E4p2JnFrRcmtpclA9ni7C5yFqTgOaf70ZdIDhGwWZbfA59VhRUqiSUGOV7+wwVCK5h9hMkjWjbcMzOykVK8y+xqa5L/kbKkIJ0JBEBcookAum2GEsj9laHY4VuOvwPvwet+g0DD7oqFYX6UyLODdedrkdJ1gOiLx/19GQs7a/CC8YLbiPwKGKI8fIo7x5FHofp9+nQX1wBo0WO/qIenKr4X3y8LA85fneIADCnUbs/HEWtX6G3MQef5f4Urwt7XjVpqDRVI9Mjjt5HZtrewf5J+Whl/4nGvC+Q++Rv0TqcKcLmImfjt/HxLw6ja7h43qvk5p1BaFvcgLTKd1Cd+Q038+hjtJ4P8dLHS3CQvYLO567Bzrf+Ctd+djhm5+Rh2cdV2N9FQ2k+klf8dv/Xy4pn2Z8JXkbr69XoT78L0wOAPNP6FiourBx40KtvQlX6M84VS89x/Py+dOTvc+X0jEf0g6k4+2P98BWAV9hDkZdXINy7SR2NB1edw49fb5W8hcq0/g7F55dgddxk93SRO5Z6NrK2ZmB22FiE3bYEGSldaOlUosfAdcbU0SuxNWcuwjAZt92fhpSmNnSO5KCGxWN1+hk8U3FKcpu51tTxCtnWkYfoWbgO659KQJjlNOrrgaykSIwVjWgPnIy41ck4/cwrMrwX7WnQrxwEAqBqliNbASqz50OU6WNw/22BUbEMvcTHYfryddiSchLvNp91hBeejO2fNaI40jHY4Sz8FqTerEeZ4ZxDsM8nocrLZzCuBKgRHq/DzVv3wdAjYdcDcwb7Szux6nvxCHOVtL/Dv/YfmB/j7yE9RwhfWxKPmLCRXrFqhN22DEsbfu2fXgWyraPRhjvjhmMnzsG67jn4tnbCcDEHroXNw3eX/gWl+8/4zeEdWUmKISQw0hMrjE/nXhNg0NN4FNU3R2BaIFJXfwOTp03FtMnXeJHDKYhPvQXVNZ9AuhktoczLC8Tu3sI5qOmfoKbxvLt3jBiPaTPgUNRiJIT7oeBeNaIkips3JPgfnCfEwPLnWryrW4mU6eNGzItDBMYMY2WhbfhYBVVSISqN7kxWZ2BpNUCv34vCpFUoFw7b4QL+/OYn0H0/2b0eYfUNSFk9HYdqTiteeZJtHUoENy7r2rbccGy/CfvnNWB59m/c6AEah+kp6Yg6ZECbhG0UocZ0Li0BP7zlpM1A8Eg7j8aadxEZow3M1nbPX1H/wR1IjZ/iBVI1wmZEIbLqKBol66EIVV7cHK0qFCZpbRV9KgorG9z8gm+YF/ag1cIwI2YCqiRzUM/BUPYHLLz/Fv+U27FxWN/GgmUF/23rETcWYMwG7H1vLn6Wl+ChfhdgLF2N+OwdMNjZGXYgOz4Fq/XDt+yZ1io8+eJBHFj3PewwfGW/2/bLTch9Fe/dno88t4caud6iFCxsqFe48iTbCoyH4W0LQK1BwiPLsLj9PHrdcXTCbsH9C9txvC3IvzQWggrhc3KKlDJuzyeoqbqCeRFTFZ5gbYGxJMm5pc21vLP0MFvz34euo3/A13+1FjovewPU0yIwD7XS9VD4i5ernglVFLL0n9tKi7e8uC/p9mBlfCZ2GAbIA7XYkT0fcav1I062HfGFbdVuHKZFRAFSOaicHT5NQGJUAH4paWlG1auXseLJZGhghtHQKvKhgPgDzrTq8eOmJag3XRlwuPpNaKwogA7NKH/x2LDOiTo6C2X/8yye3vKQQDgDy6nf4dXeJXgyeTpg/jMMrW72naojkJigcOVJthXYD3BtW3vUPphPNqE7/XbcNPykItsN4xGVGImG4zHE2yEAABGmSURBVB2K9wLaNaZfzwiQU+QZLy9jDwwFVZlnIWaG0rMywhC3/h3nljbX8q5Mg4brLjaWY/eFZShI9uGLuDAtYmJNEvVQ+JGXq54Jtg2VaTcMdK97y4tpxes/bsb99Sb0W3s+rsDUuA8FOg3M5a+iZoTW5MgvbK54DvTaxZqlcVCZ7g58GoBDvoz5A+xc+11k5y3FjDEqqMbEYcNpBuOMJYiyD61xDu6d6djXno/42fzP8i+jrTkcP9nzX0jQ2Ibc1BrEZeVjY0Ec4M4Eaac3AbdEwC+xNvkR5C2PwBiVCmO023Aa7g7pcc4soFew8gw2206wNljsn+Bzjb1MpO87hPz4+3hfxsphW4BpLUeqdQg3AitrZmHrDxLh7gw29bQbMEn/4bCOtlNxogC/ESCnSBH0fejuaINZE4WIae6+JJVQjGvZvoEXmhOx0frVTA9aje1ut7bFNDSf6EC3O93KYjcPhoUmL6atDWN+8j94KkFj6y3kFvJbhac3ZkOD0xJ/PWXCiY5zPrZOGVg6OwAlhnwtrThSYlvjR6VFUmHVsCsDqzUL8VRlE8/ZN6EmZzbGxa1Hm21oDXwH1x5mLWPjEZ32IOJcTILWZNyNeI97TMdBk7AWlSb+MN/ryHF7LTLOmY0AWkw+PX+Dj9CIB8FnW7XVnu9gfRzXsOQ39uxhXKblsC3Xg5SDGmtDxoR3tmcgzu5Mj8iZU1WLGHSM/CWiO7IojuwEyCmSHTGXgAWdLadlTImba1KDkqxYD+apcA5RlbVlm58di4nWVtBsPP7xV5hgXfRuDfTmqwDXOpsdj/x2g6BFJsiOeioi5mm9bGULZIUoL3X0vchyNc9Ek+LlfC4hO8A6lKkxo8nnCpZzTv+Om+Ue8rU0oCR9O5q//TN0siz6TZW4o6EQ8Sv3yLTmkjMza4h1yHYWtuQucLsXwIUkr4I5u938qRSNCneSJ9u6Q0mSONy78ea/o6Ob1iyShKfMQsgpkhmwVTxzDh0nTC5SYtBTt2FgFVzhFzbCc+0G1DlNZOaGv3Yh/dk2fPtnRrDsFZiOLUBD9nKsLG0YptWpRtjsLEHLdqC1rea+smBfQJpmLBxa2yy/ReYiO1IEjypeAxPKY7dkej2fSwrk4jI4Zx4yD/lya1HtwpGH8/GkrQdNrbkT2RmJgOElvN4g3Vd04nm0h3LzxN7Eibz1WOF27479Xol+Fe1RINtKZDU3xHAfP0DinmA3kqUoXhEgp8grbFLepEZ48laYhF/XiJ2btiJZ2K3PzVPZ0IiHn861zY8YB43uYWSkAIbS/WhwcqKk1F1ElrlN5hZRaPFiut7DSycexNYV0ZJPwPd5KNPqnPaKGFnCIKYDx197B7W5c6zzcAY+vb8GMblvADBKNEdtZH2Zrrew+dBt2LXGz2sxfdUu8/NjY0G2HblQSBqjV4LhbEkVImEuCJBT5AKMLMGxUZjhYh6Dt+kxbR/itdqXkRtzzdAXZmPmILfWDEg02dZb3Xy+L9R5cQvnbX4PC3flupzf4jPDoBAwASllJ22Tz/lzcliYtifLP5RlaUbF7tNYseW7mC3x8xkU+GVVkmwrK14SLjkBcookR+oPgQ+hrOVL3qRTe8XSiO3JU/2hUICnGQi8uP2mynBiRT6yA2EvMb9a7BKaPviLm2s1SawotxHq3uOI+METSPZk8qzEaoSuOLJt6No2NHNGTpGSdhX91NfXOUVcBk7ig+YvfPzSSEkQbqYVsry4Bf5exuGIx7DJl2UQ3MQY0NG49XkeToS5fB0eLRJ8ccZ8DoPhb/KVa+vO8C+iY8lKB4eIMddht176/eMC2g5yKEe2lYMqyZSZADlFMgMeWbxvc2TUUQvwcMo/UJ7+PRQJVkZmuv4IQ7vCK6nKvuxAsPPiHKLd+HlHIlbzHSKugt59wI2tA0YuUfYYmnkBuqWMXUHr73hEr1iPYh1g2JGJeO3Xh4aBZ2zH+Zt8WDvLIR3BCcf7RzlYtHkzcudMGkrTur5QAbomT4EaF2AsuReqpJ3KfgUnUDV4T8m2wWu70as5OUVK2N7+ubocaamjsWJrPnS2lZG13EJ2tq/WZjxjwk03BuBKxCNxCFlenEO0DSsX5WNHrn0ZBJu9xszAo10TB/bF4z5RT4pCUslwXw+OBFGC68zf8L7xQ7zfIbNjHZaAvJcPoro4Exqb2prMUtQatiLN0/3M3Mr2OdQVLcOibbXisUdaHqGnDoVa+2TwNwbm86WW++7QKsWby7VSaZFtAVxGx/sfwvi+jL2e4iWZQr0hwPL+AG5aCv25Q8AzVl+yLWUPscBDbFnLl+6I9zDOFdbUWM0WZ85lOb2AuWxm8e/Zlt5+D+X4EL3/JFuWomGRUsa2iCRLvPrZi8eKWI3VPpyNhP9xbMGxswMG6K1ni3WRrK64nu3lm+TiMbZAI7hPhHd/SxmbAg2bUnaSFZrCIzt89Tab/c0INrvmn3wt6FguAj7yJtvKZRhf5f6TrcmOYL+Z/Tb7lZeiPLKtl2mM1tuEbN3avcUbZ4vu4RMY2I9KgzZ+oITH3MrIaVhfyf1LKNYLUdIM2YQiL/uw39aRqYYlYP07bVgvjBmejO0mFtuF4aLnWt/32VPfiDvjFgARQdjbKMokwAOV5K1kWgGOXX71xiPizgWIw42SL7shv+6jLwUaPlPE5mqEx9+NDI3UWzkoorx7iVhMaGnSIiP1Fgk+oR4FvNyj6kUsbvuGNjSNNATkhWS6hQgQASIQ6gTIKVLKwuG3IDXj6yG7gBe3ueQJSLdVBUKcl3zFbmDfOHi1f5d8WpFkIkAEiEAwECCnSDErTUF86l1o93k/KsUU9iChgd6Jdkkr4lDm5QFaj6Ny2zdckqjHzo3ErZOOhyb32yf5O//Go7DunBsCKUrAECDbBowpSBHlCJBTpBjrgSGh9OqjaFR66w3Z88j1TpxHuiRDZ3ZlQ5mXPY8y/HLbN3x6u2Sby46ooW2eEyu2LY1DGC0kOiLLQItAtg00i5A+ChAgp0gByINJhC9A7obzqGlUapPLwZTlPbB8gjc/uAO5OolXzw5VXrJZg4Hlz7X4IG1lAG4uK1umSTARIAJEQDIC5BRJhtIdQdxiZjm49qVDvq9p4k5yisTpQ1dtHcZ8fxmiJS9NochLRqMwn6O2ehy+L8PmsnJo7TzE5s4wXHDHkYNjIMocbbYNRBuQTt4RkLwa806NUXRXWDzW5I/HoYNn5Nu+QEGcTNdh7DmTjNVxk+VJNcR4yQOJk9qHrv2v4Ez6I8puLuvDvJORh9zse/iFzq989pdBMtlWZD9J8bIoA30S6ScCtE6R4uDVCJv9AB7pOQxD+/VIjgziNWCYv8FQdz0efzIBYbJxDCFesjECmPbjqJuRhSflck5d6e7R2kmuhFB4QBIg2wakWUgpeQmQUyQvXxfSx0GTsGxwSwMXkQI/WH0jkjNuVEDPEOElIyl1ZDIyImVMgEQTASJABEYBARo+GwVGpiwSASJABIgAESACIxMgp2hkRhSDCBABIkAEiAARGAUEyCkaBUamLBIBIkAEiAARIAIjEyCnaGRGFIMIjE4CzN/wvvFDvN9xeXTmX+lcK8lbybSU5hhw6V1Gx/sfwvj+30Lii+OAwyuxQuQUSQyUxBEBIhBABBgzjJWFSFLZ1jdKKkSl0exUOTFdeuRqY5GrD42lMgLIAvKpYmnFkZIsaK22jUVWSQ1aLYxjeswZ6HNjoc3Vo0twyTEinRGBAQLkFFFJIAJEIEQJXICxdDXis3fAYM+hYQey41Ow2sH5YWBpacRhczMO17fDYo9Lv4FLgDmDA3uPY9KqvTCxV2CqXwOU3gPdZgN6+Fpb2lF/uBnmw41oETpM/Hh0TARsBMgpoqJABIhASBJgWvX4cdMS1JuuDCzC129CY0UBdGhG+YvH0EY9B0FqdwY9f/4M/746GwmacQC4JTty8fSWh2CuCsW9JYPUTEGqNq1TFKSGI7WJABEYjsBltDWH4yd7sodW+FZrEJeVj41/OQZDVRs6LQyiw7l2oRphM6IQy60cFqOVcSHS4fSla+4TUCM87k7cJnKDZkk8YsJ4bf0wLWJiNQCiMIMfLnIvBREBjgA5RVQOiAARECegvhF3xi0AIoJx1fXxiE57UDxfADQZdyPe6hANRFFPi8A8jRaImApeleryflkuKMlbybRkgSUQynyB5rYE/GbLvZjON6B6KiLmaaFBBKbxwwW3y3s6HhF3LkAcbvRf2ZI3gyEl3W/FJKQoUmaIABEIDgI9n6Cmaha25C5AeHBoTFoOS6APZqMeJf+ZjQPX3oE513PDafRHBLwnQD1F3rOjO4lAaBOwtrInoiNkctmHrqNv4kTeemyMFvR+WYdZYoAZ8u3i5xbGr0UiYpoCFXtI2PYc6grvwaIdxgG0+xJxpLkaf9qbxustCsOMmBjEwt/DohMxz5+9kG4VPorEEaCeIioHRIAIuCDAVShAS2dofI/FdL2FzYduw6418eLzhpRySFzQhsWEFkQoNPclFGw7FcnbG8GyF9FyrAwFOg3M5Vux23BOQNjfDokFnS1AjL8dbgEVOhUnQE6ROBcKJQJEAOMwLeI6fNpxzmldn6CDY2lGxe7TWLHlu5gtNuGW6zlZdLNCDok4Paa7A5/erNTclxCyLcIRnZyDbS9uQQqMqKr5hPdZPpfPOHzLnw4Jcw4dn16nTA+geNGiUA8IkFPkASyKSgRGFwHuq6wIoMUU3Gv3MF2o23scET94AsnWT7jFrDgVyU89jGi/vREZWDo7FPz6LURsyzOlOmoBHk6JRKzDF4RqhCevRpZwuJR3n+yHivYAyp6bkE/Ab6+AkCdLGSQCIUCAq2jSLn6O7mBd04dziDa9iI4lKx0cIsZch9361gDqAetDdweQlhih2JyGoLet8PninI/2e/BY6izFGApVEDtnuj/HxbQFiKLaVgxPwIWRmQLOJKQQEQggAuoIJCa043hbEO5/xjlEP8rBos2bkTtnElT2rT5UKozRFqBr8pTBytPv23wwHTjeEInEKMEEcDmLQtDalptgHQ9tVgn09i1bLKegf/YFnNuRj+XTeRPV/b7Nx2W0HW9HgoLOrpxFZjTIJqdoNFiZ8kgEvCYwHlGJkfjgzU+CbAjtHOqKlmHRtlrxnGtSkBo/Rfya4qEMLH+uxQcJ8xXuTQhW24ZjTup9iNmXj/R4LcaoYpH1wseYnPu/KEubOejoKm5GsQQtn+DNDxR2dsX0oDC3CahYlmXtsbmWFO/UHky/IgSIlQiUYYKI1zBwFLzklR241vb3nwc2bkYavxWuoN6hndQ51G0oxYW1m3ziS7YNtFLCoKduGwouPIrnfXTWvLJtoOEIUH2EbKmnKEANRWoRgYAhoJ6J5XnfwqFXTgRZb1HAEBxGEQYW429xNP4xx2GfYe6Q9BLZVlKcDsIsjSg7OhcblwdY75WDknQiJEBOkZAInRMBIuBEQB39APKnHMZe4wWnaxTgAwFLI/ZWa7DWjxUn2dYH+7m89QKMe+swc+0S3kKSLiPThQAiQE5RABmDVCECgUsgHLOz/z9uPbAH+q6+wFUzmDSzNKN883u49XF/V5xkW2mLTQ9OlZfiwK0Z/un9kzYzo04azSny0uTCcUgvxYya24hXYJjaZzswZjRU1cJ8xwNYFk27h3ltVUszKl9owJxVjyLB5dpJnkkn23rGS57YPThVWYbaOY9gXYJGsknfPttWnsyGhFQhW3KKvDSrEKSXYkbNbcQrMEwtjR24TTiP4g+d1+OO++Kgof5mD4wrHzuyrQdmkCEqYzbi4B++wIw77kacRI6uXU1pbGuXRr98AkK25BTx6XhwLATpwa2jMirxCgyzkx0Cww5yaEG2lYNqYMgk28pnByFbauPJx5okEwEiQASIABEgAkFEgJyiIDIWqUoEiAARIAJEgAjIR8Bp+Ey+pEgyESACRIAIEAEiQAQCiwB/0eqxQtX4F4XX6HyIgHAccugKHYkRIF5iVJQPIzsoz1ypFMm2SpFWPh2yrXzMObb8v/8D1XvLbAt6ngMAAAAASUVORK5CYII="> <em><strong>A1</strong></em><em><strong>A1</strong></em><em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for each correct probability for <em>Y</em> = 1, 2, 3, 4. Accept 0.162 for P(<em>Y</em> = 4).</p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</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( Y \right) = \sum {y{\text{P}}\left( {Y = y} \right)} ">
<mrow>
<mtext>E</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mi>Y</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mo>∑</mo>
<mrow>
<mi>y</mi>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>Y</mi>
<mo>=</mo>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 1 \times 0.257 \ldots + 2 \times 0.270 \ldots + 3 \times 0.189 + 4 \times 0.161 \ldots ">
<mo>=</mo>
<mn>1</mn>
<mo>×</mo>
<mn>0.257</mn>
<mo>…</mo>
<mo>+</mo>
<mn>2</mn>
<mo>×</mo>
<mn>0.270</mn>
<mo>…</mo>
<mo>+</mo>
<mn>3</mn>
<mo>×</mo>
<mn>0.189</mn>
<mo>+</mo>
<mn>4</mn>
<mo>×</mo>
<mn>0.161</mn>
<mo>…</mo>
</math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 2.01">
<mo>=</mo>
<mn>2.01</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="T">
<mi>T</mi>
</math></span> be the no of days per year that Steffi does not visit</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="T \sim B\left( {365{\text{,}}\,\,0.122 \ldots } \right)">
<mi>T</mi>
<mo>∼</mo>
<mi>B</mi>
<mrow>
<mo>(</mo>
<mrow>
<mn>365</mn>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mn>0.122</mn>
<mo>…</mo>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(M1)</strong></em></p>
<p>require <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0.45 \leqslant {\text{P}}\left( {T \leqslant n} \right) < 0.55">
<mn>0.45</mn>
<mo>⩽</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>T</mi>
<mo>⩽</mo>
<mi>n</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo><</mo>
<mn>0.55</mn>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {T \leqslant 44} \right) = 0.51">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>T</mi>
<mo>⩽</mo>
<mn>44</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0.51</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 44">
<mi>n</mi>
<mo>=</mo>
<mn>44</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="V">
<mi>V</mi>
</math></span> be the discrete random variable “number of times Steffi is not fed per day”</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{E}}\left( V \right) = 1 \times {\text{P}}\left( {X = 5} \right) + 2 \times {\text{P}}\left( {X = 6} \right) + 3 \times {\text{P}}\left( {X = 7} \right) + \ldots ">
<mrow>
<mtext>E</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mi>V</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>1</mn>
<mo>×</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo>=</mo>
<mn>5</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mn>2</mn>
<mo>×</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo>=</mo>
<mn>6</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mn>3</mn>
<mo>×</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo>=</mo>
<mn>7</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mo>…</mo>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 1 \times 0.0416 \ldots + 2 \times 0.0145 \ldots + 3 \times 0.00437 \ldots + \ldots ">
<mo>=</mo>
<mn>1</mn>
<mo>×</mo>
<mn>0.0416</mn>
<mo>…</mo>
<mo>+</mo>
<mn>2</mn>
<mo>×</mo>
<mn>0.0145</mn>
<mo>…</mo>
<mo>+</mo>
<mn>3</mn>
<mo>×</mo>
<mn>0.00437</mn>
<mo>…</mo>
<mo>+</mo>
<mo>…</mo>
</math></span> <em><strong>A1</strong></em></p>
<p>= 0.083979... <em><strong>A1</strong></em></p>
<p>expected no of occasions per year > 0.083979... × 365 = 30.7 <em><strong>A1</strong></em></p>
<p>hence Steffi can expect not to be fed on at least 30 occasions <em><strong>AG</strong></em></p>
<p><strong>Note:</strong> Candidates may consider summing more than three terms in their calculation for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{E}}\left( V \right)">
<mrow>
<mtext>E</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mi>V</mi>
<mo>)</mo>
</mrow>
</math></span>.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{E}}\left( X \right) - {\text{E}}\left( Y \right) = 0.0903 \ldots ">
<mrow>
<mtext>E</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mi>X</mi>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mrow>
<mtext>E</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mi>Y</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0.0903</mn>
<mo>…</mo>
</math></span> <em><strong>M1A</strong></em><em><strong>1</strong></em></p>
<p>0.0903… × 365 <em><strong>M1</strong></em></p>
<p>= 33.0 > 30 <em><strong> A1AG</strong></em></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">e.</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>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="T \sim {\text{N}}\left( {11.6{\text{, }}{{0.8}^2}} \right)"> <mi>T</mi> <mo>∼</mo> <mrow> <mtext>N</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mn>11.6</mn> <mrow> <mtext>, </mtext> </mrow> <mrow> <msup> <mrow> <mn>0.8</mn> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {T < \left. {10.7} \right|T < 11} \right)"> <mrow> <mtext>P</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>T</mi> <mo><</mo> <mrow> <mo stretchy="true" symmetric="true" fence="true"></mo> <mrow> <mn>10.7</mn> </mrow> <mo>|</mo> </mrow> <mi>T</mi> <mo><</mo> <mn>11</mn> </mrow> <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{{{\text{P}}\left( {T < 10.7 \cap T < 11} \right)}}{{{\text{P}}\left( {T < 11} \right)}}"> <mo>=</mo> <mfrac> <mrow> <mrow> <mtext>P</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>T</mi> <mo><</mo> <mn>10.7</mn> <mo>∩</mo> <mi>T</mi> <mo><</mo> <mn>11</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mrow> <mtext>P</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>T</mi> <mo><</mo> <mn>11</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( {T < 10.7} \right)}}{{{\text{P}}\left( {T < 11} \right)}}"> <mo>=</mo> <mfrac> <mrow> <mrow> <mtext>P</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>T</mi> <mo><</mo> <mn>10.7</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mrow> <mtext>P</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>T</mi> <mo><</mo> <mn>11</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="{\text{P}}\left( {T < 10.7} \right) = 0.1302 \ldots "> <mrow> <mtext>P</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>T</mi> <mo><</mo> <mn>10.7</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0.1302</mn> <mo>…</mo> </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( {T < 11} \right) = 0.2266 \ldots "> <mrow> <mtext>P</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>T</mi> <mo><</mo> <mn>11</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0.2266</mn> <mo>…</mo> </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( {T < \left. {10.7} \right|T < 11} \right) = 0.575"> <mrow> <mtext>P</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>T</mi> <mo><</mo> <mrow> <mo stretchy="true" symmetric="true" fence="true"></mo> <mrow> <mn>10.7</mn> </mrow> <mo>|</mo> </mrow> <mi>T</mi> <mo><</mo> <mn>11</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0.575</mn> </math></span> <em><strong> A1</strong></em></p>
<p><strong>Note:</strong> Accept only 0.575.</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>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>
<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}}(X = x + 1) = \frac{{{\mu ^{x + 1}}}}{{(x + 1)!}}{{\text{e}}^{ - \mu }}">
<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>
<mrow>
<mrow>
<msup>
<mi>μ</mi>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>!</mo>
</mrow>
</mfrac>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mi>μ</mi>
</mrow>
</msup>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{\mu }{{x + 1}} \times \frac{{{\mu ^x}}}{{x!}}{{\text{e}}^{ - \mu }}">
<mo>=</mo>
<mfrac>
<mi>μ</mi>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mo>×</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>μ</mi>
<mi>x</mi>
</msup>
</mrow>
</mrow>
<mrow>
<mi>x</mi>
<mo>!</mo>
</mrow>
</mfrac>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mi>μ</mi>
</mrow>
</msup>
</mrow>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{\mu }{{x + 1}} \times {\text{P}}(X = x)">
<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>
</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="\frac{\mu }{{x + 1}} \times {\text{P}}(X = x) = \frac{\mu }{{x + 1}} \times \frac{{{\mu ^x}}}{{x!}}{{\text{e}}^{ - \mu }}">
<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>
<mfrac>
<mi>μ</mi>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mo>×</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>μ</mi>
<mi>x</mi>
</msup>
</mrow>
</mrow>
<mrow>
<mi>x</mi>
<mo>!</mo>
</mrow>
</mfrac>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mi>μ</mi>
</mrow>
</msup>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{{\mu ^{x + 1}}}}{{(x + 1)!}}{{\text{e}}^{ - \mu }}">
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>μ</mi>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>!</mo>
</mrow>
</mfrac>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mi>μ</mi>
</mrow>
</msup>
</mrow>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{P}}(X = x + 1)">
<mo>=</mo>
<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>
</math></span> <strong><em>AG</em></strong></p>
<p><strong>METHOD 3</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{P}}(X = x + 1)}}{{{\text{P}}(X = x)}} = \frac{{\frac{{{\mu ^{x + 1}}}}{{(x + 1)!}}{{\text{e}}^{ - \mu }}}}{{\frac{{{\mu ^x}}}{{x!}}{{\text{e}}^{ - \mu }}}}">
<mfrac>
<mrow>
<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>
</mrow>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>=</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>μ</mi>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>!</mo>
</mrow>
</mfrac>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mi>μ</mi>
</mrow>
</msup>
</mrow>
</mrow>
<mrow>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>μ</mi>
<mi>x</mi>
</msup>
</mrow>
</mrow>
<mrow>
<mi>x</mi>
<mo>!</mo>
</mrow>
</mfrac>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mi>μ</mi>
</mrow>
</msup>
</mrow>
</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{{{\mu ^{x + 1}}}}{{{\mu ^x}}} \times \frac{{x!}}{{(x + 1)!}}">
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>μ</mi>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</mrow>
<mrow>
<mrow>
<msup>
<mi>μ</mi>
<mi>x</mi>
</msup>
</mrow>
</mrow>
</mfrac>
<mo>×</mo>
<mfrac>
<mrow>
<mi>x</mi>
<mo>!</mo>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>!</mo>
</mrow>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{\mu }{{x + 1}}">
<mo>=</mo>
<mfrac>
<mi>μ</mi>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p>and so <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)">
<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>
</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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(X = 3) = \frac{\mu }{3} \bullet {\text{P}}(X = 2){\text{ }}\left( {0.112777 = \frac{\mu }{3} \bullet 0.241667} \right)">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>=</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mi>μ</mi>
<mn>3</mn>
</mfrac>
<mo>∙</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>=</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>0.112777</mn>
<mo>=</mo>
<mfrac>
<mi>μ</mi>
<mn>3</mn>
</mfrac>
<mo>∙</mo>
<mn>0.241667</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p>attempting to solve for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu ">
<mi>μ</mi>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu = 1.40">
<mi>μ</mi>
<mo>=</mo>
<mn>1.40</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[3 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="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>
<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><strong>METHOD 1</strong></p>
<p>let <em>p</em> have no pets, <em>q</em> have one pet and <em>r</em> have two pets <em><strong> (M1)</strong></em></p>
<p><em>p</em> + <em>q</em> + <em>r</em> + 2 = 25 <em><strong>(A1)</strong></em></p>
<p>0<em>p</em> + 1<em>q</em> + 2<em>r</em> + 6 = 18 <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Accept a statement that there are a total of 12 pets.</p>
<p>attempt to use variance equation, or evidence of trial and error <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{0p + 1q + 4r + 18}}{{25}} - {\left( {\frac{{18}}{{25}}} \right)^2} = {\left( {\frac{{24}}{{25}}} \right)^2}">
<mfrac>
<mrow>
<mn>0</mn>
<mi>p</mi>
<mo>+</mo>
<mn>1</mn>
<mi>q</mi>
<mo>+</mo>
<mn>4</mn>
<mi>r</mi>
<mo>+</mo>
<mn>18</mn>
</mrow>
<mrow>
<mn>25</mn>
</mrow>
</mfrac>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>18</mn>
</mrow>
<mrow>
<mn>25</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>24</mn>
</mrow>
<mrow>
<mn>25</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span> <em><strong>(A1)</strong></em></p>
<p>attempt to solve a system of linear equations <em><strong>(M1)</strong></em></p>
<p><em>p</em> = 14 <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><img src="data:image/png;base64,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"> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p + q + r + \frac{2}{{25}} = 1">
<mi>p</mi>
<mo>+</mo>
<mi>q</mi>
<mo>+</mo>
<mi>r</mi>
<mo>+</mo>
<mfrac>
<mn>2</mn>
<mrow>
<mn>25</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mn>1</mn>
</math></span> <em><strong> (A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q + 2r + \frac{6}{{25}} = \frac{{18}}{{25}}\left( { \Rightarrow q + 2r = \frac{{12}}{{25}}} \right)">
<mi>q</mi>
<mo>+</mo>
<mn>2</mn>
<mi>r</mi>
<mo>+</mo>
<mfrac>
<mn>6</mn>
<mrow>
<mn>25</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mn>18</mn>
</mrow>
<mrow>
<mn>25</mn>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mo stretchy="false">⇒</mo>
<mi>q</mi>
<mo>+</mo>
<mn>2</mn>
<mi>r</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>12</mn>
</mrow>
<mrow>
<mn>25</mn>
</mrow>
</mfrac>
</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="q + 4r + \frac{{18}}{{25}} - {\left( {\frac{{18}}{{25}}} \right)^2} = \frac{{576}}{{625}}\left( { \Rightarrow q + 4r = \frac{{18}}{{25}}} \right)">
<mi>q</mi>
<mo>+</mo>
<mn>4</mn>
<mi>r</mi>
<mo>+</mo>
<mfrac>
<mrow>
<mn>18</mn>
</mrow>
<mrow>
<mn>25</mn>
</mrow>
</mfrac>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>18</mn>
</mrow>
<mrow>
<mn>25</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>576</mn>
</mrow>
<mrow>
<mn>625</mn>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mo stretchy="false">⇒</mo>
<mi>q</mi>
<mo>+</mo>
<mn>4</mn>
<mi>r</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>18</mn>
</mrow>
<mrow>
<mn>25</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(M1)(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q = \frac{6}{{25}},\,\,r = \frac{3}{{25}}">
<mi>q</mi>
<mo>=</mo>
<mfrac>
<mn>6</mn>
<mrow>
<mn>25</mn>
</mrow>
</mfrac>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>r</mi>
<mo>=</mo>
<mfrac>
<mn>3</mn>
<mrow>
<mn>25</mn>
</mrow>
</mfrac>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p = \frac{{14}}{{25}}">
<mi>p</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>14</mn>
</mrow>
<mrow>
<mn>25</mn>
</mrow>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p>so 14 have no pets</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>