THIRD EDITION A2 Pure Mathematics Catherine Berry Val Hanrahan Roger Porkess Peter Secker Series Editor: Roger Porkess Acknowledgements Wearegratefultothefollowingcompanies,institutionsandindividuals whohavegivenpermissiontoreproducephotographsinthisbook. Everyefforthasbeenmadetotraceandacknowledgeownershipof copyright.Thepublisherswillbegladtomakesuitablearrangements withanycopyrightholderswhomithasnotbeenpossibletocontact. OCR,AQAandEdexcelacceptnoresponsibilitywhatsoeverfortheaccuracyor methodofworkingintheanswersgiven. Hachette’sPolicyistousepapersthatarenatural,renewableandrecyclable productsandmadefromwoodgrowninsustainableforests.Theloggingand manufacturingprocessesareexpectedtoconformtotheenvironmentalregulations ofthecountryoforigin. 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Telephone:(44)01235827720,Fax:(44)01235400454.Linesareopenfrom9amto5pm, MondaytoSaturday,witha24hourmessageansweringservice.Youcanalsoorder throughourwebsiteatwww.hoddereducation.co.uk BritishLibraryCataloguinginPublicationData AcataloguerecordforthistitleisavailablefromtheTheBritishLibrary ISBN:978-0-340-88851-3 FirstEditionPublished1995 SecondEditionPublished2000 ThirdEditionPublished2004 Impressionnumber 10 9 Year 2011 Copyright©1995,2000,2004CatherineBerry,ValHanrahan,RogerPorkess,PeterSecker Allrightsreserved.Nopartofthispublicationmaybereproducedortransmittedinany formorbyanymeans,electronicormechanical,includingphotocopy,recording,orany informationstorageandretrievalsystem,withoutpermissioninwritingfromthepublisher orunderlicencefromtheCopyrightLicensingAgencyLimited.Furtherdetailsofsuch licences(forreprographicreproduction)maybeobtainedfromtheCopyrightLicensing AgencyLimited,ofSaffronHouse,6–10KirbyStreet,LondonEC1N8TS. TypesetbyPantekArtsLtd,Maidstone,Kent. PrintedinGreatBritainforHodderEducation,anHachetteUKcompany, 338EustonRoad,LondonNW13BHbyMPGBooksGroup. Some figures in the printed version of this book are not available for inclusion in the eBook for copyright reasons. MEI Structured Mathematics Mathematics is not only a beautiful and exciting subject in its own right but also one that underpins many other branches of learning. It is consequently fundamental to the success of a modern economy. MEI Structured Mathematics is designed to increase substantially the number of people taking the subject post-GCSE, by making it accessible, interesting and relevant to a wide range of students. It is a credit accumulation scheme based on 45 hour modules which may be taken individually or aggregated to give Advanced Subsidiary (AS) and Advanced GCE (A Level) qualifications in Mathematics, Further Mathematics and related subjects (like Statistics). The modules may also be used to obtain credit towards other types of qualification. The course is examined by OCR (previously the Oxford and Cambridge Schools Examination Board) with examinations held in January and June each year. MEI Structured Mathematics NM NC FP1 FP2 FP3 AM C2 C3 G C C1 C4 DE H S E E M1 M2 M3 M4 S1 S2 S3 S4 FAM D2 C, FP -Pure mathematics D -Decision mathematics D1 M -Mechanics N -Numerical analysis S -Statistics DE -Differential Equations DC FSMQs Additional mathematics, Foundations of Advanced Mathematics This is one of the series of books written to support the course. Its position within the whole scheme can be seen in the diagram above. iii MathematicsinEducationandIndustryisacurriculumdevelopmentbodywhich aimstopromotethelinksbetweenEducationandIndustryinMathematicsat secondarylevel,andtoproducerelevantexaminationandteachingsyllabusesand supportmaterial.Sinceitsfoundationinthe1960s,MEIhasprovidedsyllabusesfor GCSE(orOLevel),AdditionalMathematicsandALevel. FormoreinformationaboutMEIStructuredMathematicsorothersyllabusesand materials,writetoMEIOffice,AlbionHouse,MarketPlace,Westbury,Wiltshire, BA133DEorvisitwww.mei.org.uk. iv Introduction The twelve chapters of this book cover the pure mathematics required for the A2 subject criteria. The material is divided into the two units (or modules) for MEI Structured Mathematics: C3, Methods for Advanced Mathematicsand C4, Applications of Advanced Mathematics. It is the second in a series of pure mathematics books for AS and A Levels in Mathematics and Further Mathematics. Since their total content is the same, this book also covers the requirements of all the other specifications for A2 Mathematics, and it is also suitable for other courses at this level. Throughout the series the emphasis is on understanding rather than mere routine calculations, but the varied exercises do nonetheless provide plenty of scope for practising basic techniques. Extensive on-line support is available via the MEI site, www.mei.org.uk. This book is part of the third edition of this series and is written on the assumption that you have already studied AS Mathematics. Much of its content was previously in Pure Mathematics 2and3but it has now been reorganised to meet the requirements of the new specification being first taught in September 2004. Thanks are due to Val Hanrahan for her work in preparing the new edition and for her original contributions. Thanks are also due to the various examination boards who have given permission for their past questions to be included in the exercises. Roger Porkess Series Editor v Key to symbols in this book ● ? This symbol means that you may want to discuss a point with your teacher. If you are working on your own there are answers in the back of the book. It is important, however, that you have a go at answering the questions before looking up the answers if you are to understand the mathematics fully. ! This is a warning sign. It is used where a common mistake, misunderstanding or tricky point is being described. This is the ICT icon. It indicates where you should use a graphic calculator or a computer. ● This symbol invites you to join in a discussion about proof.The answers to these questions are given in the back of the book. ● This symbol and a dotted line down the right-hand side of the page indicates material which is beyond the criteria for the unit but which is included for completeness. Harder questions are indicated with stars. Many of these go beyond the usual examination standard. vi Contents C3 Methods for Advanced Mathematics 1 1 Proof 2 Proof by direct argument 2 Proof by exhaustion 3 Proof by contradiction 4 Disproof by the use of a counter-example 6 2 Natural logarithms and exponentials 8 A new function 9 The natural logarithm function 11 The exponential function 12 3 Functions 19 The language of functions 19 Using transformations to sketch the curves of functions 25 Composite functions 36 Inverse functions 39 Inverse trigonometrical functions 45 Even, odd and periodic functions 49 The modulus function 56 Curve sketching 60 4 Techniques for differentiation 63 The chain rule 63 The product rule 68 The quotient rule 71 Differentiating an inverse function 77 Differentiating natural logarithms and exponentials 82 Differentiating sinx and cosx 91 Differentiating functions defined implicitly 96 vii 5 Techniques for integration 103 Integration by substitution 103 Integrals involving the exponential function 110 Integrals involving the natural logarithm function 111 Extending the domain for logarithmic integrals 113 Integrating sinx and cosx 123 Integration by parts 125 6 Numerical solution of equations 135 Interval estimation – change of sign methods 136 Fixed point iteration 143 Rearranging the equation f(x) = 0 into the form x = g(x) 143 The Newton–Raphson method 150 C4 Applications of Advanced Mathematics 155 7 Algebra 156 The general binomial expansion 157 Review of algebraic fractions 166 Partial fractions 173 8 Trigonometry 183 Reciprocal trigonometrical functions 184 Compound-angle formulae 187 Double-angle formulae 192 The factor formulae 197 The forms r cos(θ ± α), r sin(θ ± α) 201 Small-angle approximations 210 The general solutions of trigonometrical equations 218 Using trigonometrical identities in integration 220 9 Parametric equations 224 Graphs from parametric equations 226 Finding the equation by eliminating the parameter 227 The parametric equation of a circle 231 The parametric equations of other standard curves 232 Parametric differentiation 238 viii 10 Further techniques for integration 253 Finding volumes by integration 254 The use of partial fractions in integration 261 General integration 266 Integrals you cannot do 269 11 Vectors 275 Vectors 275 Co-ordinate geometry using vectors: two dimensions 289 Co-ordinate geometry using vectors: three dimensions 303 12 Differential equations 335 Forming differential equations from rates of change 336 Solving differential equations 341 Answers 358 Chapter 1 358 Chapter 2 358 Chapter 3 359 Chapter 4 369 Chapter 5 376 Chapter 6 380 Chapter 7 383 Chapter 8 386 Chapter 9 392 Chapter 10 397 Chapter 11 403 Chapter 12 410 Index 413 ix