Haese Mathematics specialists in mathematics publishing Endorsed by Cambridge International Examinations Cambridge Additional Mathematics IGCSE® (0606) O Level (4037) Michael Haese Sandra Haese Mark Humphries Chris Sangwin 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 IGCSE01 cyan magenta yellow black Y:\HAESE\CAM4037\CamAdd_00\001CamAdd_00.cdr Tuesday, 1 April 2014 4:59:45 PM BRIAN CAMBRIDGEADDITIONALMATHEMATICS(0606)(4037) MichaelHaese B.Sc.(Hons.),Ph.D. SandraHaese B.Sc. MarkHumphries B.Sc.(Hons.) ChrisSangwin M.A.,M.Sc.,Ph.D. HaeseMathematics 152RichmondRoad,Marleston, SA5033,AUSTRALIA Telephone: +618 82104666, Fax: +618 83541238 Email: [email protected] Web: www.haesemathematics.com.au NationalLibraryofAustraliaCardNumber&ISBN 978-1-921972-42-3 ©Haese&HarrisPublications2014 PublishedbyHaeseMathematics 152RichmondRoad,Marleston, SA5033,AUSTRALIA FirstEdition 2014 CartoonartworkbyJohnMartin.CoverdesignbyBrianHouston. ArtworkbyBrianHoustonandGregoryOlesinski. FractalartworkonthecovergeneratedusingChaosPro, http://www.chaospro.de/ ComputersoftwarebyAdrianBlackburn,AshvinNarayanan,TimLee,LindenMay,SethPink,William Pietsch,andNicoleSzymanczyk. ProductionworkbyKatieRicher,GregoryOlesinski,andAnnaRijken. TypesetinAustraliabyDeanneGallaschandCharlotteFrost.TypesetinTimesRoman10. ThistextbookanditsaccompanyingCDhavebeenendorsedbyCambridgeInternationalExaminations. PrintedinChinabyProlongPressLimited. This book is copyright. 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 Mathematics. Copyingforeducationalpurposes:WherecopiesofpartorthewholeofthebookaremadeunderPartVB oftheCopyrightAct,thelawrequiresthattheeducationalinstitutionorthebodythatadministersithasgiven a remuneration notice to CopyrightAgency Limited (CAL). For information, contact the CopyrightAgency Limited. Acknowledgements: The publishers acknowledge the cooperation of Oxford University Press, Australia, for the reproduction of material originally published in textbooks produced in association with HaeseMathematics. Whileeveryattempthasbeenmadetotraceandacknowledgecopyright,theauthorsandpublishersapologisefor any accidental infringement where copyright has proved untraceable. They would be pleased to come to a suitableagreementwiththerightfulowner. Disclaimer:Alltheinternetaddresses(URLs)giveninthisbookwerevalidatthetimeofprinting.Whilethe authorsandpublisherregretanyinconveniencethatchangesof address maycausereaders,no responsibility foranysuchchangescanbeacceptedbyeithertheauthorsorthepublisher. ®IGCSEistheregisteredtrademarkofCambridgeInternationalExaminations 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 IGCSE01 cyan magenta yellow black Y:\HAESE\CAM4037\CamAdd_00\002CamAdd_00.cdr Monday, 14 April 2014 4:48:46 PM BRIAN FOREWORD This book has been written to cover the ‘Cambridge O Level Additional Mathematics (4037)’ and the ‘CambridgeIGCSE®AdditionalMathematics(0606)’coursesoveraone-yearperiod. These syllabuses enable learners to extend the mathematics skills, knowledge, and understanding developed in the Cambridge IGCSE or O Level Mathematics courses, and use skills in the context of more advanced techniques. ThesyllabuseshaveaPureMathematicsonlycontentwhichenableslearnerstoacquireasuitablefoundation in mathematics for further study in the subject. Knowledge of the content of the Cambridge IGCSE or O LevelMathematicssyllabus(oranequivalentsyllabus)isassumed. Learnerswhosuccessfullycompletethesecoursesgainlifelongskills,including: • thefurtherdevelopmentofmathematicalconceptsandprinciples (cid:129) anabilitytosolveproblems,presentsolutionslogically,andinterpretresults. This book is an attempt to cover, in one volume, the content outlined in the Cambridge O LevelAdditional Mathematics (4037) and Cambridge IGCSE Additional Mathematics (0606) syllabuses. The book can be usedasapreparationforGCEAdvancedLevelMathematics. ThebookhasbeenendorsedbyCambridge. Toreflecttheprinciplesonwhichthecourseisbased,wehaveattemptedtoproduceabookandCDpackage that embraces understanding and problem solving in order to give students different learning experiences. Reviewexercisesappearattheendofeachchapter. Answersaregivenattheendofthebook,followedbyan index. The interactive CD contains Self Tutor software (see p. 5), geometry and graphics software, demonstrations and simulations. The CD also contains the text of the book so that students can load it on a homecomputerandkeepthetextbookatschool. The examinations for Cambridge Additional Mathematics are in the form of two papers. Many of the problems in this textbook have been written to reflect the style of the examination questions.The questions, workedsolutionsandcommentsthatappearinthebookandCDwerewrittenbytheauthors. Thebookcanbeusedasaschemeofworkbutitisexpectedthattheteacherwillchoosetheorderoftopics. Exercises in the book range from routine practice and consolidation of basic skills, to problem solving exercisesthatarequitedemanding. Inthischangingworldofmathematicseducation,webelievethatthecontextualapproachshowninthisbook will enhance the students’ understanding, knowledge and appreciation of mathematics, and its universal application. Wewelcomeyourfeedback. Email: [email protected] Web: www.haesemathematics.com.au PMH,SHH,MH,CS 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 IGCSE01 cyan magenta yellow black Y:\HAESE\CAM4037\CamAdd_00\003CamAdd_00.cdr Tuesday, 8 April 2014 10:17:11 AM BRIAN ABOUT THE AUTHORS Michael Haese completed a BSc at the University of Adelaide, majoring in Infection and Immunity, and Applied Mathematics. He then completed Honours in Applied Mathematics, and a PhD in high speed fluid flows. He has a keen interest in education and a desire to see mathematics come alive in the classroom through its history and relationship with other subject areas. Michael has been the principal editor for Haese Mathematics since 2008. SandraHaesecompletedaBScattheUniversityofAdelaide,majoringinPureMathematics and Statistics. She taught mathematics at Underdale High School and later at Westminster School in Adelaide. In 1979, Sandra’s husband Bob Haese began to write textbooks for mathematics students at high schools, and Sandra assumed the role of proof reader. She continues to work for Haese Mathematics as an editor and proof reader, and she produces muchoftheaudioworkfortheSelfTutorsoftware.In2007shewasawardedLifeMembership of the Mathematics Association of South Australia. Mark Humphries completed an honours degree in Pure Mathematics and an Economics degree at the University of Adelaide. His mathematical interests include public key cryptography and number theory. He has been working at Haese Mathematics since 2006. ChrisSangwincompletedaBAinMathematicsattheUniversityofOxford,andanMScand PhD in Mathematics at the University of Bath. He spent thirteen years in the Mathematics Department at the University of Birmingham,and from 2000 - 2011 was seconded half time to the UK Higher Education Academy “Maths Stats and OR Network” to promote learning and teaching of university mathematics. He was awarded a National Teaching Fellowship in 2006, and is now a Senior Lecturer in MathematicsEducation in the MathematicsEducation Centre at Loughborough University. His learning and teaching interests include automatic assessment of mathematics using computer algebra, and problem solving using the Moore method and similar student-centredapproaches. 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 IGCSE01 cyan magenta yellow black Y:\HAESE\CAM4037\CamAdd_00\004CamAdd_00.cdr Tuesday, 8 April 2014 10:18:39 AM BRIAN USING THE INTERACTIVE CD TheinteractiveStudentCDthatcomeswiththisbookisdesignedforthosewho wThaentCtoDuitciloisnettheacthanpoploeagrysitnhtreoaucghhinogutatnhdelebaoronkindgeMnoattehseamnaaticctsiv.elinkontheCD. demonstrationsH(cid:129)garapehinsgseoftwaMre(cid:129)asimtuhlatieonsm(cid:129)reavistioincgsames(cid:129)printouts IINNTTEERRAACCTTIIVVEESSTTUUDDEENNTTCCDD SimplyclickontheiconwhenrunningtheCDtoaccessalargerangeofinteractive featuresthatincludes: includSeselfTutor www.haesemathematics.com.au EEInnCxtadaemmornribnasrateiidtodigonebnasly (cid:129) printableworksheets CaAmbdriddgeitional IGCSE®(0606), OLevel(4037) (cid:129) graphingpackages Mathematics INTERACTIVELINK (cid:129) demonstrations ©2014 (cid:129) simulations (cid:129) revisiongames (cid:129) SELFTUTOR SELF TUTOR is an exciting feature of this book. The Self Tutor icon on each worked example denotes an active link on the CD. Simply ‘click’ on the Self Tutor (or anywhere in the example box) to access the worked example, with a teacher’s voice explaining each step necessary to reach the answer. Play any line as often as you like. See how the basic processes come alive using movement and colour on the screen. Ideal for students who have missed lessons or need extra help. Example 10 Self Tutor Find the two angles µ on the unit circle, with 06µ 62¼, such that: a cosµ = 1 b sinµ = 3 c tanµ =2 3 4 a cos¡1(1)¼1:23 b sin¡1(3)¼0:848 c tan¡1(2)¼1:11 3 4 y y y 1 1 1 Er -1 O 1 x -1 O 1 x -1 O 1 x Qe -1 -1 -1 ) µ ¼1:23 or 2¼¡1:23 ) µ ¼0:848 or ¼¡0:848 ) µ ¼1:11 or ¼+1:11 ) µ ¼1:23 or 5:05 ) µ ¼0:848 or 2:29 ) µ ¼1:11 or 4:25 SeeChapter8,Theunitcircleandradianmeasure,page209 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 IGCSE01 cyan magenta yellow black Y:\HAESE\CAM4037\CamAdd_00\005CamAdd_00.cdr Wednesday, 2 April 2014 11:39:53 AM BRIAN 6 SYMBOLS AND NOTATION USED IN THIS BOOK N the set of natural numbers, f1, 2, 3, ....g Z the set of integers, f0, §1, §2, §3, ....g Z+ the set of positive integers, f1, 2, 3, ....g Q the set of rational numbers Q + the set of positive rational numbers, fx2Q , x>0g R the set of real numbers R+ the set of positive real numbers, fx2R, x>0g [a, b] the closed interval fx2R :a6x6bg [a, b) the interval fx2R :a6x<bg (a, b] the interval fx2R :a<x6bg (a, b) the open interval fx2R :a<x<bg fx , x , ....g the set with elements x , x , .... 1 2 1 2 n(A) the number of elementsin the finite set A fx: .... the set of all x such that 2 is an element of 2= is not an elementof ? or f g the empty set U the universal set [ union \ intersection µ is a subset of ½ is a proper subset of * is not a subset of 6½ is not a proper subset of A0 the complementof the set A an1, pna a to the power of 1, nth root of a (if a>0 then pna>0) n 1 p p a2, a a to the power 1, square root of a (if a>0 then a>0) 2 ½ x for x>0, x2R jxj the modulusor absolutevalue of x, that is ¡x for x<0, x2R ´ identity or is equivalentto ¼ is approximatelyequal to n! n factorialfor n2N (0!=1) ¡ ¢ n the binomialcoefficient n! for n, r2N, 06r6n r r!(n¡r)! > is greater than ¸ or > is greater than or equal to < is less than · or 6 is less than or equal to 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 IGCSE01 cyan magenta yellow black Y:\HAESE\CAM4037\CamAdd_00\006CamAdd_00.cdr Friday, 31 January 2014 11:57:28 AM BRIAN 7 Pn ai a1+a2+::::+an i=1 f functionf f : x7!y f is a functionunder which x is mapped to y f(x) the image of x under the function f f¡1 the inverse functionof the functionf g±f, gf the compositefunctionof f and g lim f(x) the limit of f(x) as x tends to a x!a dy the derivativeof y with respect to x dx d2y the second derivativeof y with respect to x dx2 f0(x) the derivativeof f(x) with respect to x f00(x) the second derivativeof f(x) with respect to x R y dx the indefiniteintegral of y with respect to x R b y dx the definite integral of y with respect to x for values of x betweena and b a e base of natural logarithms ex exponentialfunctionof x lgx logarithmof x to base 10 lnx natural logarithm of x log x logarithmto the base a of x a sin, cos, tan, the circular functions cosec, sec, cot A(x, y) the point A in the plane with Cartesian coordinatesx and y 8 <the line segmentwith endpoints A and B AB the distance from A to B : the line containingpoints A and B b A the angle at A b CAB the angle between CA and AB 4ABC the triangle whose vertices are A, B, and C a the vector a ¡! AB the vector representedin magnitudeand directionby the directed line segment from A to B jaj the magnitudeof vector a ¡! ¡! jABj the magnitudeof AB i, j unit vectors in the directionsof the Cartesian coordinateaxes M a matrix M M¡1 the inverse of the square matrix M detM the determinantof the square matrix M 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 IGCSE01 cyan magenta yellow black Y:\HAESE\CAM4037\CamAdd_00\007CamAdd_00.cdr Friday, 31 January 2014 2:46:05 PM BRIAN 8 Table of contents TABLE OF CONTENTS G Exponentialfunctions 118 H Thenaturalexponentialex 123 Reviewset4A 125 Reviewset4B 127 SYMBOLSANDNOTATION 5 LOGARITHMS 129 USEDINTHISBOOK 6 A Logarithmsinbase10 130 B Logarithmsinbasea 133 1 SETSANDVENNDIAGRAMS 11 C Lawsoflogarithms 135 A Sets 12 D Logarithmicequations 138 B Intervalnotation 15 E Naturallogarithms 142 C Relations 17 F Solvingexponentialequations D Complementsofsets 18 usinglogarithms 145 E Propertiesofunionandintersection 20 G Thechangeofbaserule 147 F Venndiagrams 21 H Graphsoflogarithmicfunctions 149 G Numbersinregions 26 Reviewset5A 152 H ProblemsolvingwithVenndiagrams 28 Reviewset5B 154 Reviewset1A 31 Reviewset1B 33 6 POLYNOMIALS 155 A Realpolynomials 156 2 FUNCTIONS 35 B Zeros,roots,andfactors 162 A Relationsandfunctions 36 C TheRemaindertheorem 167 B Functionnotation 40 D TheFactortheorem 169 C Domainandrange 43 Reviewset6A 173 D Themodulusfunction 46 Reviewset6B 173 E Compositefunctions 49 F Signdiagrams 51 7 STRAIGHTLINEGRAPHS 175 G Inversefunctions 54 A Equationsofstraightlines 177 Reviewset2A 60 B Intersectionofstraightlines 183 Reviewset2B 61 C Intersectionofastraightlineandacurve 186 D Transformingrelationshipsto 3 QUADRATICS 63 straightlineform 187 A Quadraticequations 65 E Findingrelationshipsfromdata 192 B Quadraticinequalities 72 Reviewset7A 197 C Thediscriminantofaquadratic 73 Reviewset7B 199 D Quadraticfunctions 75 E Findingaquadraticfromitsgraph 87 8 THEUNITCIRCLEANDRADIAN F Wherefunctionsmeet 91 MEASURE 201 G Problemsolvingwithquadratics 93 A Radianmeasure 202 H Quadraticoptimisation 95 B Arclengthandsectorarea 205 Reviewset3A 98 C Theunitcircleandthetrigonometricratios 208 Reviewset3B 99 D Applicationsoftheunitcircle 213 E Multiplesof "y and "r 217 4 SURDS,INDICES,AND F Reciprocaltrigonometricratios 221 EXPONENTIALS 101 Reviewset8A 221 A Surds 102 Reviewset8B 222 B Indices 107 C Indexlaws 108 9 TRIGONOMETRICFUNCTIONS 225 D Rationalindices 111 A Periodicbehaviour 226 E Algebraicexpansionandfactorisation 113 B Thesinefunction 230 F Exponentialequations 116 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 IGCSE01 cyan magenta yellow black Y:\HAESE\CAM4037\CamAdd_00\008CamAdd_00.cdr Friday, 31 January 2014 11:53:47 AM BRIAN Table of contents 9 C Thecosinefunction 236 I Derivativesofexponentialfunctions 355 D Thetangentfunction 238 J Derivativesoflogarithmicfunctions 359 E Trigonometricequations 240 K Derivativesoftrigonometricfunctions 361 F Trigonometricrelationships 246 L Secondderivatives 363 G Trigonometricequationsinquadraticform 250 Reviewset13A 365 Reviewset9A 251 Reviewset13B 366 Reviewset9B 252 14 APPLICATIONSOFDIFFERENTIAL 10 COUNTINGANDTHEBINOMIAL CALCULUS 367 EXPANSION 255 A Tangentsandnormals 369 A Theproductprinciple 256 B Stationarypoints 375 B Countingpaths 258 C Kinematics 380 C Factorialnotation 259 D Ratesofchange 388 D Permutations 262 E Optimisation 393 E Combinations 267 F Relatedrates 399 F Binomialexpansions 270 Reviewset14A 402 G TheBinomialTheorem 273 Reviewset14B 405 Reviewset10A 277 Reviewset10B 278 15 INTEGRATION 409 A Theareaunderacurve 410 11 VECTORS 279 B Antidifferentiation 415 A Vectorsandscalars 280 C Thefundamentaltheoremofcalculus 417 B Themagnitudeofavector 284 D Integration 422 C Operationswithplanevectors 285 E Rulesforintegration 424 D Thevectorbetweentwopoints 289 F Integrating f(ax+b) 428 E Parallelism 292 G Definiteintegrals 431 F Problemsinvolvingvectoroperations 294 Reviewset15A 434 G Lines 296 Reviewset15B 435 H Constantvelocityproblems 298 Reviewset11A 302 16 APPLICATIONSOFINTEGRATION 437 Reviewset11B 303 A Theareaunderacurve 438 B Theareabetweentwofunctions 440 12 MATRICES 305 C Kinematics 444 A Matrixstructure 307 Reviewset16A 449 B Matrixoperationsanddefinitions 309 Reviewset16B 450 C Matrixmultiplication 315 D Theinverseofa2×2matrix 323 ANSWERS 453 E Simultaneouslinearequations 328 Reviewset12A 330 INDEX 503 Reviewset12B 331 13 INTRODUCTIONTODIFFERENTIAL CALCULUS 333 A Limits 335 B Ratesofchange 336 C Thederivativefunction 340 D Differentiationfromfirstprinciples 342 E Simplerulesofdifferentiation 344 F Thechainrule 348 G Theproductrule 351 H Thequotientrule 353 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 IGCSE01 cyan magenta yellow black Y:\HAESE\CAM4037\CamAdd_00\009CamAdd_00.cdr Tuesday, 8 April 2014 10:19:12 AM BRIAN 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 IGCSE01 cyan magenta yellow black Y:\HAESE\CAM4037\CamAdd_00\010CamAdd_00.cdr Monday, 16 December 2013 10:41:33 AM BRIAN