Cambridge IGCSE and O Level Additional Mathematics Coursebook Second edition
University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314 321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi 110025, India 79 Anson Road, 06-04/06, Singapore 079906 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: cambridge.org/9781108411660 Cambridge University Press 2018 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 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Printed in Malaysia by Vivar Printing A catalogue record for this publication is available from the British Library ISBN 9781108411660 Paperback ISBN 9781108411738 Cambridge Elevate Edition ISBN 9781108411745 Paperback + Cambridge Elevate Edition 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 a registered trademark. Past exam paper questions throughout are reproduced by permission of Cambridge Assessment International Education. Cambridge Assessment International Education 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 in this title were written by the authors. In examinations, the way marks are awarded may be different. 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 Functions 1 1.1 Mappings 2 1.2 Definition of a function 3 1.3 Composite functions 5 1.4 Modulus functions 7 1.5 Graphs of y = f(x) where f(x) is linear 10 1.6 Inverse functions 12 1.7 The graph of a function and its inverse 15 Summary 18 Examination questions 19 2 Simultaneous equations and quadratics 23 2.1 Simultaneous equations (one linear and one non-linear) 25 2.2 Maximum and minimum values of a quadratic function 28 2.3 Graphs of y = f(x) where f(x) is quadratic 34 2.4 Quadratic inequalities 37 2.5 Roots of quadratic equations 39 2.6 Intersection of a line and a curve 42 Summary 44 Examination questions 46 3 Indices and surds 49 3.1 Simplifying expressions involving indices 50 3.2 Solving equations involving indices 51 3.3 Surds 55 3.4 Multiplication, division and simplification of surds 57 3.5 Rationalising the denominator of a fraction 60 3.6 Solving equations involving surds 63 Summary 67 Examination questions 67 4 Factors and polynomials 70 4.1 Adding, subtracting and multiplying polynomials 71 4.2 Division of polynomials 73 4.3 The factor theorem 75 4.4 Cubic expressions and equations 78 4.5 The remainder theorem 82 Summary 86 Examination questions 87 5 Equations, inequalities and graphs 89 5.1 Solving equations of the type ax b = cx d 90 5.2 Solving modulus inequalities 94 5.3 Sketching graphs of cubic polynomials and their moduli 98 5.4 Solving cubic inequalities graphically 102 5.5 Solving more complex quadratic equations 103 Summary 105 Examination questions 107 6 Logarithmic and exponential functions 111 6.1 Logarithms to base 10 112 6.2 Logarithms to base a 115 6.3 The laws of logarithms 118 vi vii viii iii
Cambridge IGCSE and O Level Additional Mathematics iv 6.4 Solving logarithmic equations 120 6.5 Solving exponential equations 122 6.6 Change of base of logarithms 124 6.7 Natural logarithms 126 6.8 Practical applications of exponential equations 128 6.9 The graphs of simple logarithmic and exponential functions 129 6.10 The graphs of y = k e nx + a and y = k ln (ax + b) where n, k, a and b are integers 130 6.11 The inverse of logarithmic and exponential functions 133 Summary 134 Examination questions 135 7 Straight-line graphs 138 7.1 Problems involving length of a line and midpoint 140 7.2 Parallel and perpendicular lines 143 7.3 Equations of straight lines 145 7.4 Areas of rectilinear figures 148 7.5 Converting from a non-linear equation to linear form 151 7.6 Converting from linear form to a non-linear equation 155 7.7 Finding relationships from data 159 Summary 165 Examination questions 165 8 Circular measure 170 8.1 Circular measure 171 8.2 Length of an arc 174 8.3 Area of a sector 177 Summary 180 Examination questions 181 9 Trigonometry 186 9.1 Angles between 0 and 90 187 9.2 The general definition of an angle 190 9.3 Trigonometric ratios of general angles 192 9.4 Graphs of trigonometric functions 195 9.5 Graphs of y = f(x), where f(x) is a trigonometric function 205 9.6 Trigonometric equations 209 9.7 Trigonometric identities 214 9.8 Further trigonometric equations 216 9.9 Further trigonometric identities 218 Summary 220 Examination questions 221 10 Permutations and combinations 224 10.1 Factorial notation 225 10.2 Arrangements 226 10.3 Permutations 229 10.4 Combinations 234 Summary 237 Examination questions 238 11 Series 243 11.1 Pascal s triangle 244 11.2 The binomial theorem 249 11.3 Arithmetic progressions 252 11.4 Geometric progressions 257 11.5 Infinite geometric series 262 11.6 Further arithmetic and geometric series 267 Summary 270 Examination questions 271 iv
Contents 12 Differentiation 1 274 12.1 The gradient function 275 12.2 The chain rule 280 12.3 The product rule 282 12.4 The quotient rule 285 12.5 Tangents and normals 287 12.6 Small increments and approximations 291 12.7 Rates of change 294 12.8 Second derivatives 298 12.9 Stationary points 300 12.10 Practical maximum and minimum problems 305 Summary 310 Examination questions 311 13 Vectors 315 13.1 Further vector notation 317 13.2 Position vectors 319 13.3 Vector geometry 323 13.4 Constant velocity problems 327 Summary 331 Examination questions 335 14 Differentiation 2 336 14.1 Derivatives of exponential functions 337 14.2 Derivatives of logarithmic functions 341 14.3 Derivatives of trigonometric functions 345 14.4 Further applications of differentiation 350 Summary 356 Examination questions 357 15 Integration 361 15.1 Differentiation reversed 362 15.2 Indefinite integrals 365 15.3 Integration of functions of the form (ax + b)n 367 15.4 Integration of exponential functions 368 15.5 Integration of sine and cosine functions 370 15.6 Integration of functions of the form 1 x and 1 372 ax + b 15.7 Further indefinite integration 375 15.8 Definite integration 378 15.9 Further definite integration 383 15.10 Area under a curve 385 15.11 Area of regions bounded by a line and a curve 392 Summary 397 Examination questions 398 16 Kinematics 402 16.1 Applications of differentiation in kinematics 404 16.2 Applications of integration in kinematics 412 Summary 418 Examination questions 419 Answers 422 Index 454 v
Cambridge IGCSE and O Level Additional Mathematics Acknowledgements Past examination paper questions throughout are reproduced by permission of Cambridge Assessment International Education. Thanks to the following for permission to reproduce images: Cover artwork: Shestakovych/Shutterstock Chapter 1 Fan jianhua/shutterstock; Chapter 2 zhu difeng/shutterstock; Chapter 3 LAGUNA DESIGN/Getty Images; Chapter 4 Michael Dechev/Shutterstock; Fig. 4.1 Steve Bower/Shutterstock; Fig. 4.2 Laboko/Shutterstock; Fig. 4.3 irin-k/shutterstock; Chapter 5 zentilia/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 elfinadesign/ Shutterstock; Chapter 12 AlenKadr/Shutterstock; Chapter 13 muratart/shutterstock; Chapter 14 Neamov/Shutterstock; Chapter 15 Ahuli Labutin/Shutterstock; Chapter 16 AlexLMX/Getty vi
Introduction This highly illustrated coursebook covers the Cambridge IGCSE and O Level Additional Mathematics syllabuses (0606 and 4037). The course is aimed at students who are currently studying or have previously studied Cambridge IGCSE Mathematics (0580) or Cambridge O Level Mathematics (4024). 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. Class discussion sections have been included to 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. Challenge questions have been included at the end of most exercises to challenge and stretch highability 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. A Practice Book is also available in the Cambridge IGCSE 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. A Teacher s Resource, 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 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. Examination questions Worked example 3 2 The function f is such that f( x) = 4x 8 x + ax + b, where a and b are constants. It is given that 2x 1 is a factor of f(x) and that when f(x) is divided by x + 2 the remainder is 20. Find the remainder when f(x) is divided by x 1. [6] Answer 3 2 f( x) = 4x 8x + ax + b 1 If 2x 1 is a factor, then f = 0. 2 4 1 3 8 1 2 + 1 a + b = 0 2 2 2 1 + 1 2 a + b = 0 2 2 1 a + b = 1 1 2 2 a + 2b = 3 ----------------------(1) Cambridge IGCSE Additional Mathematics 0606 Paper 11 Q2 Nov 2011 x Remainder = 20 when divided by x + 2, means that f( 2) = 20. 3 2 4( 2) 8( 2) + a( 2) + b = 20 32 32 2a + b = 20 2a + b = 84 ---------------------(2) From (1) a = 3 2b. Substituting in (2), gives: 23 ( 2b) + b = 84 6 + 4b + b = 84 5b = 90 b = 18 So a = 33, b = 18. 3 2 Remainder when f( x) = 4x 8x 33x + 18 is divided by (x 1) is f(1). 3 2 Remainder = 41 () 81 () 33() 1 + 18 = 4 8 33 + 18 = 19 Activity for you to apply your theoretical knowledge to a practical task. y = cos x + 1 is a translation of y = cos x by the vector 0 1 y = cos x + 2 is a translation of y = cos x by the vector 0 2 0 y = cos x 3 is a translation of y = cos x by the vector 3 and y = tan x + 1 is a translation of y = tan x by the vector 0 1 0 y = tan x 2 is a translation of y = tan x by the vector 2