for the international student Mathematics HL (Options) Mathematics Peter Blythe Peter Joseph Paul Urban David Martin Robert Haese Michael Haese Specialists in mathematics publishing HAESE HARRIS PUBLICATIONS& International Baccalaureate Diploma Programme Including coverage on CD of the option forGeometry Further Mathematics SL IBHL_OPT cyan black 0 5 25 75 50 95 100 0 5 25 75 95 100 50 Y:\HAESE\IBHL_OPT\IBHLOPT_00\001IBO00.CDR Friday, 19 August 2005 9:06:19 AM PETERDELL MATHEMATICS FOR THE INTERNATIONAL STUDENT International Baccalaureate Mathematics HL (Options) This book is copyright Copying for educational purposes Acknowledgements Disclaimer Peter Blythe B.Sc. Peter Joseph M.A.(Hons.), Grad.Cert.Ed. Paul Urban B.Sc.(Hons.), B.Ec. David Martin B.A., B.Sc., M.A., M.Ed.Admin. Robert Haese B.Sc. Michael Haese B.Sc.(Hons.), Ph.D. Haese & Harris Publications 3 Frank Collopy Court, Adelaide Airport, SA 5950, AUSTRALIA Telephone: +61 8 8355 9444, Fax: + 61 8 8355 9471 Email: National Library of Australia Card Number & ISBN 1 876543 33 7 © Haese & Harris Publications 2005 Published by Raksar Nominees Pty Ltd 3 Frank Collopy Court, Adelaide Airport, SA 5950, AUSTRALIA First Edition 2005 2006 (twice) Cartoon artwork by John Martin. Artwork by Piotr Poturaj and David Purton. Cover design by Piotr Poturaj. Computer software by David Purton. Typeset in Australia by Susan Haese and Charlotte Sabel (Raksar Nominees). Typeset in Times Roman 10 /11 The textbook and its accompanying CD have been developed independently of the International Baccalaureate Organization (IBO). The textbook and CD are in no way connected with, or endorsed by, the IBO. . Except as permitted by the Copyright Act (any fair dealing for the purposes of private study, research, criticism or review), no part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publisher. Enquiries to be made to Haese & Harris Publications. : Where copies of part or the whole of the book are made under Part VB of the Copyright Act, the law requires that the educational institution or the body that administers it has given a remuneration notice to Copyright Agency Limited (CAL). For information, contact the Copyright Agency Limited. : The publishers acknowledge the cooperation of many teachers in the preparation of this book. A full list appears on page 4. While every attempt has been made to trace and acknowledge copyright, the authors and publishers apologise for any accidental infringement where copyright has proved untraceable. They would be pleased to come to a suitable agreement with the rightful owner. : All the internet addresses (URL’s) given in this book were valid at the time of printing. While the authors and publisher regret any inconvenience that changes of address may cause readers, no responsibility for any such changes can be accepted by either the authors or the publisher. Reprinted \Qw_\Qw_ info@haeseandharris.com.au www.haeseandharris.com.au Web: IBHL_OPT cyan black 0 5 25 75 50 95 100 0 5 25 75 95 100 50 R:\BOOKS\IB_books\IBHL_OPT\IBHLOPT_00\002ibo00.cdr Wednesday, 16 August 2006 10:22:48 AM PETERDELL Mathematics for the International Student: Mathematics HL (Options) Mathematics HL (Core) Further Mathematics SL has been written as a companion book to the textbook. Together, they aim to provide students and teachers with appropriate coverage of the two-year Mathematics HL Course (first examinations 2006), which is one of the courses of study in the International Baccalaureate Diploma Programme. It is not our intention to define the course. Teachers are encouraged to use other resources. We have developed the book independently of the International Baccalaureate Organization (IBO) in consultation with many experienced teachers of IB Mathematics. The text is not endorsed by the IBO. On the accompanying CD, we offer coverage of the Euclidean Geometry Option for students undertaking the IB Diploma course . This Option (with answers) can be printed from the CD. The interactive features of the CD allow immediate access to our own specially designed geometry packages, graphing packages and more. Teachers are provided with a quick and easy way to demonstrate concepts, and students can discover for themselves and re-visit when necessary. Instructions appropriate to each graphics calculator problem are on the CD and can be printed for students. These instructions are written for Texas Instruments and Casio calculators. In this changing world of mathematics education, we believe that the contextual approach shown in this book, with associated use of technology, will enhance the students understanding, knowledge and appreciation of mathematics and its universal application. We welcome your feedback Email: Web: PJB PJ PMU DCM RCH PMH info@haeseandharris.com.au www.haeseandharris.com.au FOREWORD IBHL_OPT cyan black 0 5 25 75 50 95 100 0 5 25 75 95 100 50 Y:\HAESE\IBHL_OPT\IBHLOPT_00\003IBO00.CDR Wednesday, 17 August 2005 9:09:05 AM PETERDELL The authors and publishers would like to thank all those teachers who have read the proofs of this book and offered advice and encouragement. Special thanks to Mark Willis for permission to include some of his questions in HL Topic 8 ‘Statistics and probability’. Others who offered to read and comment on the proofs include: Mark William Bannar-Martin, Nick Vonthethoff, Hans-Jørn Grann Bentzen, Isaac Youssef, Sarah Locke, Ian Fitton, Paola San Martini, Nigel Wheeler, Jeanne-Mari Neefs, Winnie Auyeungrusk, Martin McMulkin, Janet Huntley, Stephanie DeGuzman, Simon Meredith, Rupert de Smidt, Colin Jeavons, Dave Loveland, Jan Dijkstra, Clare Byrne, Peter Duggan, Jill Robinson, Sophia Anastasiadou, Carol A. Murphy, Janet Wareham, Robert Hall, Susan Palombi, Gail A. Chmura, Chuck Hoag, Ulla Dellien, Richard Alexander, Monty Winningham, Martin Breen, Leo Boissy, Peter Morris, Ian Hilditch, Susan Sinclair, Ray Chaudhuri, Graham Cramp. To anyone we may have missed, we offer our apologies. The publishers wish to make it clear that acknowledging these individuals does not imply any endorsement of this book by any of them, and all responsibility for the content rests with the authors and publishers. ACKNOWLEDGEMENTS IBHL_OPT cyan black 0 5 25 75 50 95 100 0 5 25 75 95 100 50 Y:\HAESE\IBHL_OPT\IBHLOPT_00\004IBO00.CDR Thursday, 18 August 2005 11:49:41 AM PETERDELL TABLE OF CONTENTS 5 FURTHER MATHEMATICS SL TOPIC 1 HL TOPIC 8 HL TOPIC 9 HL TOPIC 10 GEOMETRY STATISTICS AND PROBABILITY SETS, RELATIONS AND GROUPS SERIES AND DIFFERENTIAL EQUATIONS Available only by clicking on the icon alongside. This chapter plus answers is fully printable. 9 A Expectation algebra 10 B Cumulative distribution functions 19 C Distributions of the sample mean 45 D Confidence intervals for means and proportions 60 E Significance and hypothesis testing 73 F The Chi-squared distribution 88 Review set 8A 101 Review set 8B 104 109 A Sets 110 B Ordered pairs 119 C Functions 131 D Binary operations 136 E Groups 145 F Further groups 159 Review set 9A 166 Review set 9B 169 171 A Some properties of functions 174 B Sequences 190 C Infinite series 199 (Further mathematics SL Topic 2) (Further mathematics SL Topic 3) (Further mathematics SL Topic 4) TABLE OF CONTENTS TOPIC 1 IBHL_OPT cyan black 0 5 25 75 50 95 100 0 5 25 75 95 100 50 Y:\HAESE\IBHL_OPT\IBHLOPT_00\005IBO00.CDR Monday, 15 August 2005 4:45:45 PM PETERDELL 6 TABLE OF CONTENTS D Taylor and Maclaurin series E First order differential equations Review set 10A Review set 10B Review set 10C Review set 10D Review set 10E A.1 Number theory introduction A.2 Order properties and axioms A.3 Divisibility, primality and the division algorithm A.4 Gcd, lcm and the Euclidean algorithm greatest common divisor (gcd) A.5 The linear diophantine equation A.6 Prime numbers A.7 Linear congruence A.8 The Chinese remainder theorem A.9 Divisibility tests A.10 Fermat’s little theorem B.1 Preliminary problems involving graph theory B.2 Terminology B.3 Fundamental results of graph theory B.4 Journeys on graphs and their implication B.5 Planar graphs B.6 Trees and algorithms B.7 The Chinese postman problem B.8 The travelling salesman problem (TSP) Review set 11A Review set 11D Review set 11E DISCRETE MATHEMATICS A NUMBER THEORY B GRAPH THEORY Review set 11B Review set 11C 223 229 242 242 243 244 245 247 248 248 249 256 263 270 274 278 286 289 292 296 296 297 301 310 316 319 332 336 339 342 343 345 351 411 ax by c¡+¡ ¡=¡ 340 341 HL TOPIC 11 APPENDIX (Methods of proof) ANSWERS INDEX (Further mathematics SL Topic 5) IBHL_OPT cyan black 0 5 25 75 50 95 100 0 5 25 75 95 100 50 Y:\HAESE\IBHL_OPT\IBHLOPT_00\006IBO00.CDR Tuesday, 16 August 2005 10:13:45 AM PETERDELL E(X) the expected value of X, which is ¹ Var(X) the variance of X, which is ¾ 2 X Z = X ¡¹ ¾ the standardised variable P( ) the probability of occurring » is distributed as ¼ is approximately equal to x the sample mean s 2 n the sample variance s 2 n¡1 the unbiassed estimate of ¾ 2 ¹ X the mean of random variable X ¾ X the standard deviation of random variable X DU(n) the discrete uniform distribution B(n, p) the binomial distribution B(1, p) the Bernoulli distribution Hyp(n, M, N) the hypergeometric distribution Geo(p) the geometric distribution NB(r, p) the negative binomial distribution Po(m) the Poisson distribution U(a, b) the continuous uniform distribution Exp(¸) the exponential distribution N(¹, ¾ 2 ) the normal distribution bp the random variable of sample proportions X the random variable of sample means T the random variable of the t-distribution º the number of degrees of freedom H 0 the null hypothesis H 1 the alternative hypothesis  2 calc the chi-squared statistic SYMBOLS AND NOTATION f g the set of all elements 2 is an element of =2 is not an element of fx j the set of all x such that N the set of all natural numbers Z the set of integers Q the set of rational numbers R the set of real numbers C the set of all complex numbers Z + the set of positive integers P the set of all prime numbers U the universal set ; or fg the empty (null) set µ is a subset of ½ is a proper subset of P (A) the power of set A A \ B the intersection of sets A and B A [ B the union of sets A and B ) implies that )Á does not imply that A 0 the complement of the set A n(A) the number of elements in the set A A nB the difference of sets A and B A¢B the symmetric difference of sets A and B A £B the Cartesian product of sets A and B R a relation of ordered pairs xRy x is related to y x ´ y(mod n) x is equivalent to y, modulo n Z n the set of residue classes, modulo n £ n multiplication, modulo n 2Z the set of even integers f : A ! B f : x 7! yfis a function under which x is mapped to y f(x) the image of x under the function f f ¡1 the inverse function of the function f f A B is a function under which each element of set has an image in set IBHL_OPT Y:\HAESE\IBHL_OPT\IBHLOPT_00\007IBO00.CDR Monday, 15 August 2005 11:01:14 AM PETERDELL f ±g or f(g(x)) the composite function of f and g jxj the modulus or absolute value of x [ a , b ] the closed interval, a 6 x 6 b ] a, b [ the open interval a<x<b u n the nth term of a sequence or series fu n g the sequence with nth term u n S n the sum of the first n terms of a sequence S 1 the sum to infinity of a series n X i=1 u i u 1 + u 2 + u 3 + ::::: + u n n Q i=1 u i u 1 £u 2 £u 3 £::::: £u n lim x!a f(x) the limit of f(x) as x tends to a lim x!a+ f(x) the limit of f(x) as x tends to a from the positive side of a maxfa, bg the maximum value of a or b 1 X n=0 c n x n the power series whose terms have form c n x n a j badivides b, or a is a factor of b a jÁ badoes not divide b, or a is a not a factor of b gcd(a, b) the greatest common divisor of a and b lcm(a, b) the least common multiple of a and b » = is isomorphic to G is the complement of G A matrix A A n matrix A to the power of n A(G) the adjacency matrix of G A(x, y) the point A in the plane with Cartesian coordinates x and y [AB] the line segment with end points A and B AB the length of [AB] (AB) the line containing points A and B b A the angle at A [ CAB or ]CAB the angle between [CA] and [AB] ¢ABC the triangle whose vertices are A, B and C or the area of triangle ABC k is parallel to kÁ is not parallel to ? is perpendicular to AB.CD length AB £ length CD PT 2 PT £ PT Power M C the power of point M relative to circle C ¡! AB the vector from A to B IBHL_OPT cyan black 0 5 25 75 50 95 100 0 5 25 75 95 100 50 Y:\HAESE\IBHL_OPT\IBHLOPT_00\008IBO00.CDR Monday, 15 August 2005 10:59:52 AM PETERDELL Statistics and probability 88 Contents: A B C D E F Expectation algebra Cumulative distribution functions (for discrete and continuous variables) Distribution of the sample mean and the Central Limit Theorem Confidence intervals for means and proportions Significance and hypothesis testing and errors The Chi-squared distribution, the “goodness of fit” test, the test for the independence of two variables. Before beginning any work on this option, it is recommended that a careful revision of the core requirements for statistics and probability is made. This is identified by “ ” as expressed in the syl- labus guide on pages 26–29 of IBO document on the Diploma Programme Mathe- matics HL for the first examination 2006. Throughout this booklet, there will be many references to the core requirements, taken from “Mathematics for the International Student Mathematics HL (Core)” Paul Urban et al, published by Haese and Harris, especially chapters 18, 19, and 30. This will be referred to as “from the text”. Topic 6 – Core: Statistics and Probability HL Topic (Further Mathematics SL Topic 2) IBHL_OPT cyan black 0 5 25 75 50 95 100 0 5 25 75 95 100 50 Y:\HAESE\IBHL_OPT\IBHLOPT_08\009IBO08.CDR Wednesday, 17 August 2005 3:48:10 PM PETERDELL 10 STATISTICS AND PROBABILITY (Topic 8) Recall that if a random variable X has mean ¹ then ¹ is known as the expected value of X, or simply E(X). ¹ = E(X)= ( P xP (x), for discrete X R xf(x) dx, for continuous X From section 30E.1 of the text (Investigation 1) we noticed that E(aX + b)=aE(X)+b Proof: (discrete case only) E(aX + b)= P (ax + b)P (x) = P [axP( x)+bP (x)] = a P xP(x)+b P P (x) = aE(X)+b(1) fas P P (x)=1g = aE(X)+b A random variable X, has variance ¾ 2 , also known as Var(X) where ¾ 2 = Var(X) = E((X ¡ ¹) 2 ) Notice that for discrete X ² Var(X) = P (x ¡ ¹) 2 p(x) ² Var(X) = P x 2 p(x) ¡ ¹ 2 ² Var(X) = E(X 2 ) ¡fE(X)g 2 Again, from Investigation 1 of Section 30E.1, Var(aX + b) = a 2 Var(X) Proof: (discrete case only) Va r(aX + b)=E((aX + b) 2 ) ¡fE(aX + b)g 2 = E ¡ a 2 X 2 +2abX + b 2 ¢ ¡faE(X)+bg 2 = a 2 E(X 2 )+2ab E(X)+b 2 ¡a 2 fE(X)g 2 ¡2ab E(X) ¡ b 2 = a 2 E(X 2 ) ¡ a 2 fE(X)g 2 = a 2 [E(X 2 ) ¡fE(X)g 2 ] = a 2 Va r (X) The standardised variable Z is defined as Z = X ¡ ¹ ¾ and has mean 0 and variance 1. EXPECTATION ALGEBRA A E( )XX, THE EXPECTED VALUE OF Var( )XX¡ , THE VARIANCE OF THE STANDARDISED VARIABLE, Z If a random variable is normally distributed with mean and variance we write N , , where reads . X¹¾ X¹¾ 2 »»()2 is distributed as IBHL_OPT cyan black 0 5 25 75 50 95 100 0 5 25 75 95 100 50 [...]... 50 25 5 0 100 95 75 50 25 5 0 cyan black IBHL_OPT 31 STATISTICS AND PROBABILITY (Topic 8) Reminder: For the uniform distribution in Example 13 the sample space U = f1, 2, 3, 4, , ng However, the n distinct outcomes of a uniform distribution do not have to equal the set U The Mathematics HL information booklet available for tests and examinations contains the table shown below: DISCRETE DISTRIBUTIONS... 25) Find: a the mean and standard deviation of the random variable U = 3X + 2Y: b P(U < 0) 3 The marks in an IB Mathematics HL exam are distributed normally with mean ạ and standard deviation ắ If the cut off score for a 7 is a mark of 80%, and 10% of students get a 7, and the cut off score for a 6 is a mark of 65% and 30% of students get a 6 or 7, find the mean and standard deviation of the marks in... determine whether the approximation was a good one 6 The cook at a school needs to buy five dozen eggs for a school camp The eggs are sold by the dozen Being experienced the cook checks for rotten eggs He selects two eggs simultaneously from the dozen pack and if they are not rotten he purchases the dozen eggs Given that there is one rotten egg on average in each carton of one dozen eggs, find: a the probability... Calculate the probability that more than 250 ticket holders will arrive for the flight c Calculate the probability that there will be empty seats on this flight d Calculate the: i mean ii variance of X: iii Hence use a suitable approximation for X to calculate the probability that more than 250 ticket holders will arrive for the flight iv Use a suitable approximation for X to calculate the probability there... wait no longer to book a holiday The tourist decides to book for the earliest date for which the probability that snow will have fallen on or before that date is greater than 0:85 Find the exact date of the booking 8 In a board game for four players, each player must roll two fair dice in turn to get a difference of no more than 3 before they can begin to play a Find the probability of getting a difference... tournament purchase these balls They sample 2 balls from each carton and if they are both not faulty, they purchase the carton b Find the proportion of all cartons that would be rejected by the purchasers How many of 1000 cartons would the buyers expect to reject? Hint: Draw a probability distribution table for X Calculate a probability distribution for rejecting a carton for each of the values of X... inclusive a What is the mean expected score obtained on this wheel during the day? b What is the standard deviation of the scores obtained during the day? c What is the probability of getting a multiple of 7 in one spin of the wheel? If the wheel is spun 500 times during the day: d What is the likelihood of getting a multiple of 7 more than 15% of the time? Given that 20 people play each time the wheel is... these values for the mean and variance can be found using the rules for calculating mean and variance given above, the formal treatment of proofs of means and variances are excluded from the syllabus However, just as in Example 12, it is possible to derive these values In the case of the Binomial distribution, using the result that 100 95 75 50 25 5 0 100 95 75 50 25 5 0 cyan black IBHL_OPT 32 STATISTICS... Section 30H of the Core text mx eĂm where x = 0, 1, 2, 3, 4, x! and m is the mean and variance of the Poisson random variable x X mx eĂm i.e., E(X) = Var(X) = m and the cdf is F (x) = P(X 6 x) = x! r=0 It has probability mass function P(X = x) = Note: For the Poisson distribution, the mean always equals the variance We write X ằ P0 (m) to indicate that X is the random variable for the Poisson distribution,... nature were in the Core HL text Example 8 A class of IB students contains 10 females and 9 males A student committee of three is to be randomly chosen If X is the number of females on the committee, b P(X = 1) c P(X = 2) d P(X = 3) find: a P(X = 0) Ă Â 19 or C 3 The total number of unrestricted committees = 19 3 fas there are 19 students to choose from and we want any 3 of themg Ă 10  à 9  The number . 2006 10:22:48 AM PETERDELL Mathematics for the International Student: Mathematics HL (Options) Mathematics HL (Core) Further Mathematics SL has been written as. AM PETERDELL MATHEMATICS FOR THE INTERNATIONAL STUDENT International Baccalaureate Mathematics HL (Options) This book is copyright Copying for educational