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1. INTERNATIONAL BACCALAUREATEMATHEMATICSHIGHER LEVEL(CORE)i3RD EDITION3rd imprintSeries editor: Fabio CirritoContributing authors:Nigel BuckleIain Dunbar

2. MATHEMATICS Higher Level (Core)Copyright Nigel Buckle, Iain Dunbar, Key-Strokes Pty Ltd, Mifasa Pty Ltd.First published in 1997 by IBID Press2nd Edition published in 1999 by IBID Press,3rd Edition published in 2004 by IBID Press, 2nd imprint published in 2005Reprinted 2007Published by IBID Press, Victoria.Library Catalogue:Cirrito Fabio Editor., Buckle & Dunbar1. Mathematics, 2. International Baccalaureate. Series Title: InternationalBaccalaureate in DetailiiISBN: 1 876659 11 4 (10 digit)978 1 876659 11 0 (13 digit)All rights reserved except under the conditions described in the Copyright Act 1968 ofAustralia and subsequent amendments. No part of this publication may be reproduced,stored in a retrieval system, or transmitted in any form or by any means, without theprior permission of the publishers.While every care has been taken to trace and acknowledge copyright, the publisherstender their apologies for any accidental infringement where copyright has proveduntraceable. They would be pleased to come to a suitable arrangement with the rightfulowner in each case.This book has been developed independently of the International BaccalaureateOrganisation (IBO). The text is in no way connected with, or endorsed by, the IBO.This publication is independently produced for use by teachers and students. Althoughreferences have been reproduced with permission of the VCAA the publication is in noway connected or endorsed by the VCAA.We also wish to thank the Incorporated Association of Registered Teachers of Victoria forgranting us permission to reproduced questions from their October Examination Papers.Cover design by Adcore Creative.Published by IBID Press, at www.ibid.com.auFor further information contact fabio@ibid.com.auPrinted by SHANNON Books, Australia. 3. PREFACE TO 3RD EDITIONIt will be immediately obvious that the 3rd edition of the Mathematics Higher Level (Core) texthas been completely revised and updated. Sections of the previous two editions are still present,but much has happened to improve the text both in content and accuracy.In response to the many requests and suggestions from teachers worldwide the text wasextensively revised. There are more examples for students to refer to when learning the subjectmatter for the first time or during their revision period. There is an abundance of well-gradedexercises for students to hone their skills. Of course, it is not expected that any one student workthrough every question in this text - such a task would be quite a feat. It is hoped then thatteachers will guide the students as to which questions to attempt. The questions serve to developroutine skills, reinforce concepts introduced in the topic and develop skills in making appropriateuse of the graphics calculator.The text has been written in a conversational style so that students will find that they are notsimply making reference to an encyclopedia filled with mathematical facts, but rather find thatthey are in some way participating in or listening in on a discussion of the subject matter.Throughout the text the subject matter is presented using graphical, numerical, algebraic andverbal means whenever appropriate. Classical approaches have been judiciously combined withmodern approaches reflecting new technology - in particular the use of the graphics calculator.The book has been specifically written to meet the demands of the Higher Level (Core) section ofthe course and has been pitched at a level that is appropriate for students studying this subject.The book presents an extensive coverage of the syllabus and in some areas goes beyond what isrequired of the student. Again, this is for the teacher to decide how best to use these sections.Sets of revision exercises are included in the text. Many of the questions in these sets have beenaimed at a level that is on par with what a student could expect on an examination paper.However, some of the questions do go beyond the level that students may expect to find on anexamination paper. Success in examinations will depend on an individuals preparation and theywill find that making use of a selection of questions from a number of sources will be verybeneficial.I hope that most of the suggestions and recommendations that were brought forward have beenaddressed in this edition. However, there is always room for improvement. As always, I welcomeand encourage teachers and students to contact me with feedback, not only on their likes anddislikes but suggestions on how the book can be improved as well as where errors and misprintsoccur. There will be updates on the IBID Press website in relation to errors that have beenlocated in the book so we suggest that you visit the IBID website at www.ibid.com.au. If youbelieve you have located an error or misprint please email me at fabio@ibid.com.au.Fabio Cirrito, July 2004iii 4. MATHEMATICS Higher Level (Core)PREFACE TO 2ND EDITIONWe are grateful to all those teachers who have made comments and corrections on thefirst edition. We hope that these contributions have improved this second edition. Thisedition is now in line with the course whose first examinations will start in 2000. Asalways, we welcome all comments from teachers and with due time, will make use ofthem to further improve this book. Suggestions and comments can be directed to FabioCirrito via email: fabio@ibid.com.auFabio Cirrito, 1999PREFACE TO 1ST EDITIONThis text has been produced independently as a resource to support the teaching of theMathematics Higher Level Course of the International Baccalaureate. The examples andquestions do not necessarily reflect the views of the official senior examining teamappointed by the International Baccalaureate Organisation.The notation used is, as far as possible, that specified in the appropriate syllabusguidelines of the IB.The units of physical measurements are in S.I.The language and spelling are U.K. English.Currency quantities are specified in dollars, though these could be read as any currencythat is decimalised, such as Swiss francs, Lire etc.The graphic calculators covered directly in the text are the Texas TI/82 and 83.Supplementary material is available from the publisher for students using some othermakes and models of calculators. As it is important that students learn to interpretgraphic calculator output, the text and answers present a mixture of graphic calculatorscreens and conventional diagrams when discussing graphs.The text has been presented in the order in which the topics appear in the syllabus. Thisdoes not mean that the topics have to be treated in this order, though it is generally thecase that the more fundamental topics appear at the front of the book. Students arereminded that it is the IB Syllabus that specifies the contents of the course and not thistext. One of the keys to success in this course is to be thoroughly familiar with the coursecontents and the styles of questions that have been used in past examinations.Fabio Cirrito, August 1997.iv 5. CONTENTS1 THEORY OF KNOWLEDGE 11.1 Pure and Applied Mathematics 11.2 Axioms 21.3 Proof 41.3.1 Rules of Inference 51.3.2 Proof by Exhaustion 61.3.3 Direct Proof 71.3.4 Proof by Contradiction 81.4 Paradox 101.4.1 What is a Paradox? 101.4.2 Russells Paradox? 111.5 Mathematics and Other Disciplines 121.6 The nded say15 Exte Es2 ALGEBRA OF LINEAR AND QUADRATIC EXPRESSIONS 172.1 The Real Number Line 172.1.1 The Real Number Line 172.1.2 Set Builder Notation 172.1.3 Interval Notation 172.1.4 Number Systems 192.1.5 Irrational Numbers 202.1.6 The Absolute Value 222.2 Linear Algebra 242.2.1 Review of Linear Equations 242.2.2 Linear Inequations 292.3 Linear Functions 332.3.1 Graph of the Linear Function 332.3.2 Simultaneous Linear Equations in Two Unknowns 372.3.3 Simultaneous Linear Equations in Three Unknowns 422.4 Quadratics 452.4.1 Quadratic Equation 452.4.2 Quadratic Function 512.4.3 Quadratic Inequalities 582.4.4 Simultaneous Equations Involving Linear-Quadratic Equations 623 POLYNOMIALS 673.1 Algebra of Polynomials 673.1.1 Definition 673.1.2 Addition and Multiplication of Polynomials 673.1.3 Division of Polynomials 683.2 Synthetic Division 723.3 The Remainder Theorem 753.4 The Factor Theorem 773.5 Equations and Inequations 833.5.1 Polynomial Equations 833.5.2 Polynomial Inequations 863.6 Sketching Polynomials 883.6.1 Graphical Significance of Roots 883.6.2 Cubic Functions 89v 6. MATHEMATICS Higher Level (Core)4 THE BINOMIAL THEOREM 954.1 The Binomial Theorem 954.1.1 The Binomial Theorem 954.1.2 The General term 1004.2 Proof 1045 FUNCTIONS AND RELATIONS 1055.1 Relations 1055.1.1 Relations 1055.1.2 The Cartesian Plane 1075.1.3 Implied Domain 1085.1.4 Types of Relations 1095.1.5 Sketching with the Graphics Calculator 1105.2 Functions 1155.2.1 Definitions 1155.3 Some Standard Functions 1225.3.1 Hybrid Functions and Continuity 1225.3.2 The Absolute Value Function 1265.3.3 The Exponential Function 1315.3.4 The Logarithmic Function 1385.3.5 Equations of the Form y = xn, n = 1, 21445.4 Algebra of Functions 1485.4.1 Basic Operations and Composite Functions 1485.4.2 Identity and Inverse Functions 1576 TRANSFORMATIONS OF GRAPHS 1676.1 Translations 1676.1.1 Horizontal Translation 1676.1.2 Vertical Translation 1706.2 Dilations 1776.2.1 Dilation from the x-axis 1776.2.2 Dilation from the y-axis 1796.3 Reflections 1836.4 Reciprocal of a Function 1887 EXPONENTIAL AND LOGARITHMIC FUNCTIONS 1977.1 Exponents 1977.1.1 Basic Rules of Indices 1977.1.2 Indicial Equations 2017.1.3 Equations of the form b f (x) = bg(x)2037.1.4 What if the Base is Not The Same? 2057.1.5 A Special Base (e) 2077.2 Exponential Modelling 2097.3 Logarithms 2177.3.1 What are Logarithms 2177.3.2 Can we find the Logarithm of a Negative Number? 2197.4 The Algebra of Logarithms 2217.5 Logarithmic Modelling 230REVISIO