Cambridge IGCSE and O Level Additional Mathematics Coursebook

Cambridge IGCSE and O Level Additional Mathematics Coursebook

University Printing House, Cambridge CB2 8BS, United Kingdom Cambridge University Press is part of the University of Cambridge. It furthers the University s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. Information on this title: education.cambridge.org Cambridge University Press 2016 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2016 Printed in India by Multivista Global Pvt. Ltd A catalogue record for this publication is available from the British Library isbn 978-1-316-60564-6 Paperback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Information regarding prices, travel timetables, and other factual information given in this work is correct at the time of first printing but Cambridge University Press does not guarantee the accuracy of such information thereafter. IGCSE is the registered trademark of Cambridge International Examinations. Past exam paper questions throughout are reproduced by permission of Cambridge International Examinations. Cambridge International Examinations bears no responsibility for the example answers to questions taken from its past question papers which are contained in this publication. All exam-style questions and sample answers have been written by the authors. notice to teachers in the uk It is illegal to reproduce any part of this work in material form (including photocopying and electronic storage) except under the following circumstances: (i) where you are abiding by a licence granted to your school or institution by the Copyright Licensing Agency; (ii) where no such licence exists, or where you wish to exceed the terms of a licence, and you have gained the written permission of Cambridge University Press; (iii) where you are allowed to reproduce without permission under the provisions of Chapter 3 of the Copyright, Designs and Patents Act 1988, which covers, for example, the reproduction of short passages within certain types of educational anthology and reproduction for the purposes of setting examination questions.

Contents Acknowledgements Introduction How to use this book 1 Sets 1 1.1 The language of sets 2 1.2 Shading sets on Venn diagrams 6 1.3 Describing sets on a Venn diagram 9 1.4 Numbers of elements in regions on a Venn diagram 10 Summary 15 Examination questions 16 2 Functions 19 2.1 Mappings 20 2.2 Deinition of a function 21 2.3 Composite functions 23 2.4 Modulus functions 25 2.5 Graphs of y = f(x) where f(x) is linear 28 2.6 Inverse functions 30 2.7 The graph of a function and its inverse 33 Summary 36 Examination questions 37 3 Simultaneous equations and quadratics 41 3.1 Simultaneous equations (one linear and one non-linear) 43 3.2 Maximum and minimum values of a quadratic function 46 3.3 Graphs of y = f(x) where f(x) is quadratic 52 3.4 Quadratic inequalities 55 3.5 Roots of quadratic equations 57 3.6 Intersection of a line and a curve 60 Summary 62 Examination questions 64 4 Indices and surds 67 4.1 Simplifying expressions involving indices 68 4.2 Solving equations involving indices 69 4.3 Surds 73 4.4 Multiplication, division and simpliication of surds 75 4.5 Rationalising the denominator of a fraction 78 4.6 Solving equations involving surds 81 Summary 85 Examination questions 85 5 Factors and polynomials 88 5.1 Adding, subtracting and multiplying polynomials 89 5.2 Division of polynomials 91 5.3 The factor theorem 93 5.4 Cubic expressions and equations 96 5.5 The remainder theorem 100 Summary 104 Examination questions 105 6 Logarithmic and exponential functions 107 6.1 Logarithms to base 10 108 6.2 Logarithms to base a 111 6.3 The laws of logarithms 114 6.4 Solving logarithmic equations 116 vi vii viii iii

Cambridge IGCSE and O Level Additional Mathematics iv 6.5 Solving exponential equations 118 6.6 Change of base of logarithms 120 6.7 Natural logarithms 122 6.8 Practical applications of exponential equations 124 6.9 The graphs of simple logarithmic and exponential functions 125 6.10 The graphs of y = k e nx + a and y = k ln (ax + b) where n, k, a and b are integers 126 6.11 The inverse of logarithmic and exponential functions 129 Summary 130 Examination questions 131 7 Straight-line graphs 134 7.1 Problems involving length of a line and mid-point 136 7.2 Parallel and perpendicular lines 139 7.3 Equations of straight lines 141 7.4 Areas of rectilinear igures 144 7.5 Converting from a non-linear equation to linear form 147 7.6 Converting from linear form to a non-linear equation 151 7.7 Finding relationships from data 155 Summary 161 Examination questions 161 8 Circular measure 166 8.1 Circular measure 167 8.2 Length of an arc 170 8.3 Area of a sector 173 Summary 176 Examination questions 177 9 Trigonometry 180 9.1 Angles between 0 and 90 181 9.2 The general deinition of an angle 184 9.3 Trigonometric ratios of general angles 186 9.4 Graphs of trigonometric functions 189 9.5 Graphs of y = f(x), where f(x) is a trigonometric function 199 9.6 Trigonometric equations 202 9.7 Trigonometric identities 208 9.8 Further trigonometric equations 210 9.9 Further trigonometric identities 212 Summary 214 Examination questions 215 10 Permutations and combinations 218 10.1 Factorial notation 219 10.2 Arrangements 220 10.3 Permutations 223 10.4 Combinations 227 Summary 231 Examination questions 232 11 Binomial expansions 236 11.1 Pascal s triangle 237 11.2 The binomial theorem 242 Summary 245 Examination questions 245 12 Differentiation 1 247 12.1 The gradient function 248 12.2 The chain rule 253 12.3 The product rule 255 iv

Contents 12.4 The quotient rule 258 12.5 Tangents and normals 260 12.6 Small increments and approximations 264 12.7 Rates of change 267 12.8 Second derivatives 271 12.9 Stationary points 273 12.10 Practical maximum and minimum problems 278 Summary 283 Examination questions 284 13 Vectors 288 13.1 Further vector notation 290 13.2 Position vectors 292 13.3 Vector geometry 296 13.4 Constant velocity problems 300 13.5 Interception problems 304 13.6 Relative velocity 307 13.7 Relative velocity using i, j notation 314 Summary 315 Examination questions 316 14 Matrices 319 14.1 Addition, subtraction and multiplication by a scalar 320 14.2 Matrix products 322 14.3 Practical applications of matrix products 325 14.4 The inverse of a 2 2 matrix 330 14.5 Simultaneous equations 334 Summary 336 Examination questions 340 15 Differentiation 2 341 15.1 Derivatives of exponential functions 342 15.2 Derivatives of logarithmic functions 346 15.3 Derivatives of trigonometric functions 350 15.4 Further applications of differentiation 355 Summary 361 Examination questions 362 16 Integration 365 16.1 Differentiation reversed 366 16.2 Indeinite integrals 369 16.3 Integration of functions of the form (ax + b) n 371 16.4 Integration of exponential functions 372 16.5 Integration of sine and cosine functions 374 16.6 Further indeinite integration 376 16.7 Deinite integration 380 16.8 Further deinite integration 383 16.9 Area under a curve 385 16.10 Area of regions bounded by a line and a curve 391 Summary 396 Examination questions 397 17 Kinematics 400 17.1 Applications of differentiation in kinematics 402 17.2 Applications of integration in kinematics 410 Summary 416 Examination questions 417 Answers 419 Index 449 v

Cambridge IGCSE and O Level Additional Mathematics Acknowledgements Past examination paper questions throughout are reproduced by permission of Cambridge International Examinations. Thanks to the following for permission to reproduce images: Cover artwork: Shestakovych/Shutterstock Chapter 1 YuriyS/Getty Images; Chapter 2 Fan jianhua/shutterstock; Chapter 3 zhu difeng/ Shutterstock; Chapter 4 LAGUNA DESIGN/Getty Images; Fig. 4.1 Steve Bower/Shutterstock; Fig. 4.2 Laboko/Shutterstock; Fig. 4.3 irin-k/shutterstock; Chapter 5 Michael Dechev/Shutterstock; Chapter 6 Peshkova/Shutterstock; Chapter 7 ittipon/shutterstock; Chapter 8 Zhu Qiu/EyeEm/ Getty Images; Chapter 9 paul downing/getty Images; Fig. 9.1 aarrows/shutterstock; Chapter 10 Gino Santa Maria/Shutterstock; Fig. 10.1snake3d/Shutterstock; Fig. 10.2 Keith Publicover/Shutterstock; Fig. 10.3 Aleksandr Kurganov/Shutterstock; Fig. 10.4 Africa Studio/Shutterstock; Chapter 11 Ezume Images/Shutterstock; Chapter 12 AlenKadr/Shutterstock; Chapter 13 muratart/shutterstock; Fig. 13.1 Rawpixel.com/Shutterstock; Fig. 13.2 & 13.3 Michael Shake/Shutterstock; Chapter 14 nadla/ Getty Images; Chapter 15 Neamov/Shutterstock; Chapter 16 Ahuli Labutin/Shutterstock; Chapter 17 AlexLMX/Getty vi

Introduction This highly illustrated coursebook offers full coverage of the Cambridge IGCSE and O Level Additional Mathematics syllabuses (0606 and 4037). It has been written by a highly experienced author, who is very familiar with the syllabus and the examinations. The course is aimed at students who are currently studying or have previously studied Cambridge IGCSE Mathematics (0580) or Cambridge O Level Mathematics (4024). The coursebook has been written with a clear progression from start to inish, with some later chapters requiring knowledge learned in earlier chapters. Where the content in one chapter includes topics that should have already been covered in previous studies, a recap section has been provided so that students can build on their prior knowledge. At the start of each chapter, there is a list of objectives that are covered in the chapter. These objectives have been taken directly from the Cambridge IGCSE Additional Mathematics (0606) syllabus. Class discussion sections have been included. These provide students with the opportunity to discuss and learn new mathematical concepts with their classmates, with their class teacher acting as the facilitator. The aim of these class discussion sections is to improve the student s reasoning and oral communication skills. Worked examples are used throughout to demonstrate each method using typical workings and thought processes. These present the methods to the students in a practical and easy-to-follow way that minimises the need for lengthy explanations. The exercises are carefully graded. They offer plenty of practice via drill questions at the start of each exercise, which allow the student to practise methods that have just been introduced. The exercises then progress to questions that typically relect the kinds of questions that the student may encounter in the examinations. Challenge questions have also been included at the end of most exercises to challenge and stretch high-ability students. Towards the end of each chapter, there is a summary of the key concepts to help students consolidate what they have just learnt. This is followed by a Past paper questions section, which contains real questions taken from past examination papers. The answers to all questions are supplied at the back of the book, allowing self- and/or class-assessment. A student can assess their progress as they go along, choosing to do more or less practice as required. The answers given in this book are concise and it is important for students to appreciate that in the examination they should show as many steps in their working as possible. A Practice Book is also available in the Additional Mathematics series, which offers students further targeted practice. This book closely follows the chapters and topics of the coursebook, offering additional exercises to help students to consolidate concepts learnt and to assess their learning after each chapter. Clues and Tips are included to help students with tricky topics. A Teacher s resource CD-ROM, to offer support and advice, is also available. vii

Cambridge IGCSE and O Level Additional Mathematics How to use this book Chapter each chapter begins with a set of learning objectives to explain what you will learn in this chapter. viii Recap check that you are familiar with the introductory skills required for the chapter. Class Discussion additional activities to be done in the classroom for enrichment.

How to use this book Worked Example detailed step-by-step approaches to help students solve problems. Note quick suggestions to remind you about key facts and highlight important points. Challenge Q challenge yourself with tougher questions that stretch your skills. ix Summary at the end of each chapter to review what you have learnt.

Cambridge IGCSE and O Level Additional Mathematics Examination questions exam-style questions for you to test your knowledge and understanding at the end of each chapter. x