STUDENT BOOK MATHEMATICS SAMPLE - Pearson
PEARSON EDEXCEL INTERNATIONAL A LEVEL Content is fully mapped to the specification to provide comprehensive coverage andeasy referenceEngaging and relevant international content in a real-world contextExam-style questions at the end of each chapter, and an exam practice paper at theend of the book, provide practice for exam writing techniqueSignposted transferable skills prepare for further education and employmentReviewed by a language specialist to ensure the book is written in a clear andaccessible styleGlossary of key Mathematics terminology, and full answers, included at the back ofthe bookInteractive practice activities also includedIAL FURTHERPURE MATHS 1Student BookISBN: 9781292244648An Online Teacher Resource Pack (9781292244631) provides further planning, teachingand assessment support.This Student Book supports the following qualifications:Pearson Edexcel International Advanced Subsidiary in Further Mathematics (XFM01)Pearson Edexcel International Advanced Level in Further Mathematics (YFM01)For first teaching September 2018IAL FURTHERPURE MATHS 2Student BookISBN: 9781292244655www.pearsonglobalschools.comIAL FPM3 Cover.indd 1-3PLE eBookincludedPEARSON EDEXCEL INTERNATIONAL A LEVELFURTHER PUREMATHEMATICS 3MPearson Edexcel International A Level Further Pure Mathematics 3 Student Book providescomprehensive coverage of the Further Pure Mathematics 3 unit. This book is designed toprovide students with the best preparation possible for the examination:MATHEMATICSSTUDENT BOOKSASTUDENT BOOKPEARSON EDEXCEL INTERNATIONAL A LEVEL FURTHER PURE MATHEMATICS 3 STUDENT BOOKFURTHER PUREMATHEMATICS 325/06/2019 13:17
PEARSON EDEXCEL INTERNATIONAL A LEVELFURTHER PUREMATHEMATICS 3Student BookSample material. Not for resale, circulation or distribution in whole or in part. Pearson 2020.Series Editors: Joe Skrakowski and Harry SmithAuthors: Greg Attwood, Jack Barraclough, Tom Begley, Ian Bettison, Lee Cope,Alistair Macpherson, Bronwen Moran, Johnny Nicholson, Laurence Pateman,Joe Petran, Keith Pledger, Joe Skrakowski, Harry Smith, Geoff Staley,Dave Wilkins
Published by Pearson Education Limited, 80 Strand, London, WC2R 0RL.www.pearsonglobalschools.comCopies of official specifications for all Pearson qualifications may be found on thewebsite: https://qualifications.pearson.comText Pearson Education Limited 2020Edited by Linnet BruceTypeset by Tech-Set Ltd, Gateshead, UKOriginal illustrations Pearson Education Limited 2020Illustrated by Tech-Set Ltd, Gateshead, UKCover design by Pearson Education Limited 2020The rights of Greg Attwood, Jack Barraclough, Tom Begley, Ian Bettison, Lee Cope,Charles Garnet Cox, Alistair Macpherson, Bronwen Moran, Johnny Nicholson,Laurence Pateman, Joe Petran, Keith Pledger, Joe Skrakowski, Harry Smith, GeoffStaley and Dave Wilkins to be identified as the authors of this work have beenasserted by them in accordance with the Copyright, Designs and Patents Act 1988.First published 202023 22 21 2010 9 8 7 6 5 4 3 2 1British Library Cataloguing in Publication DataA catalogue record for this book is available from the British LibrarySample material. Not for resale, circulation or distribution in whole or in part. Pearson 2020.ISBN 978 1 292244 66 2Copyright noticeAll rights reserved. No part of this may be reproduced in any form or by any means(including photocopying or storing it in any medium by electronic means andwhether or not transiently or incidentally to some other use of this publication)without the written permission of the copyright owner, except in accordance withthe provisions of the Copyright, Designs and Patents Act 1988 or under the termsof a licence issued by the Copyright Licensing Agency, 5th Floor, ShackletonHouse, 4 Battlebridge Lane, London, SE1 2HX (www.cla.co.uk). Applications forthe copyright owner’s written permission should be addressed to the publisher.Printed in Slovakia by NeografiaPicture CreditsThe authors and publisher would like to thank the following individuals andorganisations for permission to reproduce photographs:Shutterstock.com: spacedrone808 46; Getty Images: MediaProduction 1,Westend61 54, Science Photo Library 100, Abstract Aerial Art 137; 123rf.com:destinacigdem 17Cover images: Front: Getty Images: Werner Van SteenInside front cover: Shutterstock.com: Dmitry LobanovAll other images Pearson Education Limited 2020All artwork Pearson Education Limited 2020Endorsement StatementIn order to ensure that this resource offers high-quality support for the associatedPearson qualification, it has been through a review process by the awarding body.This process confirms that this resource fully covers the teaching and learningcontent of the specification or part of a specification at which it is aimed. It alsoconfirms that it demonstrates an appropriate balance between the developmentof subject skills, knowledge and understanding, in addition to preparation forassessment.Endorsement does not cover any guidance on assessment activities or processes(e.g. practice questions or advice on how to answer assessment questions)included in the resource, nor does it prescribe any particular approach to theteaching or delivery of a related course.While the publishers have made every attempt to ensure that advice on thequalification and its assessment is accurate, the official specification andassociated assessment guidance materials are the only authoritative source ofinformation and should always be referred to for definitive guidance.Pearson examiners have not contributed to any sections in this resource relevant toexamination papers for which they have responsibility.Examiners will not use endorsed resources as a source of material for anyassessment set by Pearson. Endorsement of a resource does not mean that theresource is required to achieve this Pearson qualification, nor does it mean that itis the only suitable material available to support the qualification, and any resourcelists produced by the awarding body shall include this and other appropriateresources.
CONTENTSCOURSE STRUCTURE ivABOUT THIS BOOK viQUALIFICATION AND ASSESSMENT OVERVIEW Sample material. Not for resale, circulation or distribution in whole or in part. Pearson 2020.iiiviiiEXTRA ONLINE CONTENT x1 HYPERBOLIC FUNCTIONS 12 FURTHER COORDINATE SYSTEMS 173 DIFFERENTIATION 464 INTEGRATION 54REVIEW EXERCISE 1 935 VECTORS 1006 FURTHER MATRIX ALGEBRA 137REVIEW EXERCISE 2 191EXAM PRACTICE 199GLOSSARY 201ANSWERS 204INDEX 244
ivCOURSE STRUCTURECHAPTER 1 HYPERBOLICFUNCTIONS 11.1 INTRODUCTION TO HYPERBOLICFUNCTIONS 21.2 SKETCHING GRAPHS OF HYPERBOLICFUNCTIONS 41.3 INVERSE HYPERBOLIC FUNCTIONS 71.4 IDENTITIES AND EQUATIONS 10CHAPTER REVIEW 1 14Sample material. Not for resale, circulation or distribution in whole or in part. Pearson 2020.CHAPTER 2 FURTHERCOORDINATE SYSTEMS 2.1 ELLIPSES 2.2 HYPERBOLAS 2.3 ECCENTRICITY 2.4 TANGENTS AND NORMALS TOAN ELLIPSE 2.5 TANGENTS AND NORMALS TOA HYPERBOLA 2.6 L OCI CHAPTER REVIEW 2 1718202229333842CHAPTER 3 DIFFERENTIATION 463.1 DIFFERENTIATING HYPERBOLICFUNCTIONS 3.2 DIFFERENTIATING INVERSEHYPERBOLIC FUNCTIONS 3.3 DIFFERENTIATING INVERSETRIGONOMETRIC FUNCTIONS CHAPTER REVIEW 3 47495052CHAPTER 4 INTEGRATION 4.1 STANDARD INTEGRALS 4.2 INTEGRATION 4.3 TRIGONOMETRIC ANDHYPERBOLIC SUBSTITUTIONS 4.4 INTEGRATING EXPRESSIONS 4.5 INTEGRATING INVERSETRIGONOMETRIC ANDHYPERBOLIC FUNCTIONS 4.6 DERIVING AND USINGREDUCTION FORMULAE 4.7 FINDING THE LENGTH OFAN ARC OF A CURVE 4.8 FINDING THE AREA OF ASURFACE OF REVOLUTION CHAPTER REVIEW 4 54555861677173798287REVIEW EXERCISE 1 93CHAPTER 5 VECTORS 1005.1 VECTOR PRODUCT 5.2 FINDING AREAS 5.3 SCALAR TRIPLE PRODUCT 5.4 STRAIGHT LINES 5.5 VECTOR PLANES 5.6 SOLVING GEOMETRIC PROBLEMS CHAPTER REVIEW 5 101106110115117121130
COURSE STRUCTURESample material. Not for resale, circulation or distribution in whole or in part. Pearson 2020.CHAPTER 6 FURTHER MATRIXALGEBRA 1376.1 TRANSPOSING A MATRIX 6.2 THE DETERMINANT OF A3 3 MATRIX 6.3 THE INVERSE OF A 3 3 MATRIXWHERE IT EXISTS 6.4 USING MATRICES TO REPRESENTLINEAR TRANSFORMATIONS IN3 DIMENSIONS 6.5 USING INVERSE MATRICES TOREVERSE THE EFFECT OF ALINEAR TRANSFORMATION 6.6 THE EIGENVALUES ANDEIGENVECTORS OF 2 2 AND3 3 MATRICES 6.7 REDUCING A SYMMETRIC MATRIXTO DIAGONAL FORM CHAPTER REVIEW 6 138142vREVIEW EXERCISE 2 191EXAM PRACTICE 199GLOSSARY 201ANSWERS 204INDEX 244146152160165175185
viABOUT THIS BOOKABOUT THIS BOOKThe following three themes have been fully integrated throughout the Pearson Edexcel InternationalAdvanced Level in Mathematics series, so they can be applied alongside your learning.1. Mathematical argument, language and proof Rigorous and consistent approach throughout Notation boxes explain key mathematical language and symbols2. Mathematical problem-solving Hundreds of problem-solving questions, fully integratedinto the main exercises Problem-solving boxes provide tips and strategies Challenge questions provide extra stretch3. Transferable skillsThe Mathematical Problem-Solving Cyclespecify the probleminterpret resultscollect informationprocess andrepresent information Transferable skills are embedded throughout this book, in the exercises and in some examplesSample material. Not for resale, circulation or distribution in whole or in part. Pearson 2020. These skills are signposted to show students which skills they are using and developingFinding your way around the bookEach chapter is mapped to thespecification content for easyreferenceEach chapter starts with alist of Learning objectivesThe Prior knowledgecheck helps make sureyou are ready to start thechapterGlossary terms willbe identified by boldblue text on their firstappearanceThe real world applications ofthe maths you are about to learnare highlighted at the start of thechapter
ABOUT THIS BOOKStep-by-step workedexamples focus on thekey types of questionsyou’ll need to tackleExercise questionsare carefully gradedso they increase indifficulty and graduallybring you up to examstandardExercises are packedwith exam-stylequestions to ensure youare ready for the examsSample material. Not for resale, circulation or distribution in whole or in part. Pearson 2020.Exam-style questionsare flagged with EProblem-solvingquestions are flaggedwith PProblem-solving boxes provide hints,tips and strategies, and Watch outboxes highlight areas where studentsoften lose marks in their examsEach section beginswith an explanationand key learning pointsEach chapter ends with a Chapter reviewand a Summary of key pointsAfter every few chapters, a Review exercisehelps you consolidate your learning withlots of exam-style questionsA full practice paper at the back ofthe book helps you prepare for thereal thingvii
viii QUALIFICATION AND ASSESSMENT OVERVIEWQUALIFICATION ANDASSESSMENT OVERVIEWQualification and content overviewFurther Pure Mathematics 3 (FP3) is an optional* unit in the following qualifications:International Advanced Subsidiary in Further MathematicsInternational Advanced Level in Further Mathematics*It is compulsory to study either FP2 or FP3 for the International Advanced Level in FurtherMathematics.Assessment overviewThe following table gives an overview of the assessment for this unit.We recommend that you study this information closely to help ensure that you are fully prepared forthis course and know exactly what to expect in the assessment.UnitPercentageSample material. Not for resale, circulation or distribution in whole or in part. Pearson 2020.FP3: Further PureMathematics 3Paper code WFM03/01133 3 % of IASMarkTimeAvailability751 hour 30 minsJanuary and June216 3 % of IALFirst assessment June 2020IAS: International Advanced Subsidiary, IAL: International Advanced A Level.Assessment objectives and weightingsMinimumweighting inIAS and IALAO1Recall, select and use their knowledge of mathematical facts, concepts and techniques in avariety of contexts.30%AO2Construct rigorous mathematical arguments and proofs through use of precise statements,logical deduction and inference and by the manipulation of mathematical expressions,including the construction of extended arguments for handling substantial problemspresented in unstructured form.30%AO3Recall, select and use their knowledge of standard mathematical models to representsituations in the real world; recognise and understand given representations involvingstandard models; present and interpret results from such models in terms of the originalsituation, including discussion of the assumptions made and refinement of such models.10%AO4Comprehend translations of common realistic contexts into mathematics; use the results ofcalculations to make predictions, or comment on the context; and, where appropriate, readcritically and comprehend longer mathematical arguments or examples of applications.5%AO5Use contemporary calculator technology and other permitted resources (such as formulaebooklets or statistical tables) accurately and efficiently; understand when not to use suchtechnology, and its limitations. Give answers to appropriate accuracy.5%
QUALIFICATION AND ASSESSMENT OVERVIEWixRelationship of assessment objectives to unitsAssessment objectiveFP3Marks out of 133 3 –4033 3 –4026 3 19 3 –136 3 –13 3 21CalculatorsStudents may use a calculator in assessments for these qualifications. Centres are responsible formaking sure that calculators used by their students meet the requirements given in the table below.Students are expected to have available a calculator with at least the following keys: , –, , , π, x2,1yx x , x , x , ln x, e , x!, sine, cosine and tangent and their inverses in degrees and decimals of a degree,and in radians; memory.Sample material. Not for resale, circulation or distribution in whole or in part. Pearson 2020.ProhibitionsCalculators with any of the following facilities are prohibited in all examinations: databanks retrieval of text or formulae built-in symbolic algebra manipulations symbolic differentiation and/or integration language translators communication with other machines or the internet
xEXTRA ONLINE CONTENTExtra online contentWhenever you see an Online box, it means that there is extra online content available to support you.SolutionBankSolutionBank provides worked solutions for questions in the book. Downloadthe solutions as a PDF or quickly find the solution you need online.Use of technologyExplore topics in more detail, visualiseproblems and consolidate your understanding.Use pre-made GeoGebra activities or Casioresources for a graphic calculator.Sample material. Not for resale, circulation or distribution in whole or in part. Pearson 2020.GeoGebra-powered interactivesyxOnline Find the point of intersectiongraphically using technology.Graphic calculator interactivesInteract with the mathsyou are learning usingGeoGebra's easy-to-usetoolsInteract with the maths you are learningusing GeoGebra's easy-to-use toolsExplore the maths you are learning and gainconfidence in using a graphic calculatorCalculator tutorialsOur helpful video tutorials willguide you through how to useyour calculator in the exams.They cover both Casio's scientificand colour graphic calculators.Online Work out each coefficient quickly usingthe nCr and power functions on your calculator.Step-by-step guide with audio instructionson exactly which buttons to press and whatshould appear on your calculator's screen
1 HYPERBOLICFUNCTIONS1.11.21.3Learning objectivesAfter completing this chapter you should be able to: Understand the definitions of the hyperbolic functions pages 2–3Sample material. Not for resale, circulation or distribution in whole or in part. Pearson 2020. Sketch the graphs of the hyperbolic functions and know their properties pages 4–7 Understand and use inverse hyperbolic functions including theirgraphs, properties and logarithmic equivalents pages 7–10 Prove identities and solve equations using hyperbolic functions pages 10–14Prior knowledge check12f(x) 2ex e xSolve the equation f(x) 2 Show that1 tan 2 x 1 cos 2 x Pure 3 Section 4.2 Pure 2 Section 6.3Hyperbolic curves feature often inarchitectural modelling. A hanging chainmight look like a parabola but it is actuallya curve called a catenary (derived fromthe Latin word for chain). In buildings theinverted catenary provides a very stablestructure.
2CHAPTER 1HYPERBOLIC FUNCTIONS1.1 Introduction to hyperbolic functionsHyperbolic functions have several properties incommon with trigonometric functions, but theyare defined in terms of exponential functions.Notation Hyperbolic sine (or sinh) is definedex e xas sinh x 2NotationOften pronounced ‘shine’. Hyperbolic cosine (or cosh) is definedex e x as cosh x 2NotationOften pronounced ‘cosh’. Hyperbolic tangent (or tanh) is definedsinh x as tanh x cosh xNotationOften pronounced ‘tanch’ or ‘than’.x belongs to the set of real numbers,using the correct mathematical notation.You can use the definitions of sinh x and cosh x to write tanh x in exponential form.Sample material. Not for resale, circulation or distribution in whole or in part. Pearson 2020.sinh x e x e x e x e x 2 tanh x x e e x e x e x 2cosh xMultiplying the numerator and denominator of the final expression through by e x gives:e2x 1 2x tanh x e 1There are also hyperbolic functions corresponding to (i.e. connected to) the reciprocal trigonometricfunctions: Hyperbolic cosecant (or cosech) is defined12as cosech x x e e x sinh xNotationOften pronounced ‘cosech’ or ‘cosheck’. Hyperbolic secant (or sech) is defined12as sech x x e e x cosh xNotationOften pronounced ‘sheck’ or ‘setch’. Hyperbolic cotangent (or coth) is defined e 2x 11as coth x 2xtanh x e 1NotationOften pronounced ‘coth’.Example1SKILLSANALYSISFind, to 2 decimal places, the values ofa sinh 3b cosh 1 3 e 3 ea sinh 3 10.02 (2 d.p.)2 1 e 1 eb cosh 1 1.54 (2 d.p.)2 e 1.6 – 1c tanh 0.8 1.6 0.66 (2 d.p.) e 1c tanh 0.8
HYPERBOLIC FUNCTIONSExampleCHAPTER 132Find the exact value of tanh (ln 4).2e2ln4 1eln4 1eln16 1 ln16tanh (ln 4) 2ln4 2e 1 eln4 1 e 11516 1 16 1 17ExampleUse elnk k.3Use the definition of sinh x to find, to 2 decimal places, the value of x for which sinh x 5ex e x 5 ex e x 102e2x 1 10exe2x 10ex 1 0x e 5 26 e x 5 26 So x ln (5 26 ) 2.31 (2 d.p.)Sample material. Not for resale, circulation or distribution in whole or in part. Pearson 2020. Exercise1ASKILLSMultiply both sides by ex.The substitution u ex turns this into thequadratic equation u2 – 10u – 1 0ex cannot be negative.ANALYSIS1 Use your calculator to find, to 2 decimal places, the value of:1 ) c tanh ( 2)a sinh 4b cosh (22 Write, in terms of e:a sinh 1d sech 5b cosh 4c tanh 0.5d sech ( 1)b cosh (ln 3)c tanh (ln 2)d cosech (ln π)3 Find the exact values of:a sinh (ln 2)In questions 4 to 8, use the definitions of the hyperbolic functions (in terms of exponentials) to findeach answer, then check your answers using an inverse hyperbolic function on your calculator.4 Find, to 2 decimal places, the values of x for which cosh x 25 Find, to 2 decimal places, the values of x for which sinh x 116 Find, to 2 decimal places, the values of x for which tanh x 27 Find, to 2 decimal places, the values of x for which coth x 1018 Find, to 2 decimal places, the values of x for which sech x 8
4HYPERBOLIC FUNCTIONSCHAPTER 11.2 Sketching graphs of hyperbolic functionsyYou can sketch the graphs of the hyperbolic functions byconsidering the graphs of y ex and y e–xex (–e–
MATHEMATICS Pearson Edexcel International A Level Further Pure Mathematics 3 Student Book provides comprehensive coverage of the Further Pure Mathematics 3 unit. This book is designed to . 2020. Published by Pearson Education Limited, 80 Strand, London, WC2R 0RL. www.pearsonglobalschools.com
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Pearson Edexcel International Advanced Subsidiary/Advanced Level in Mathematics, Further Mathematics and Pure Mathematics Mathematical Formulae and Statistical Tables For use in Pearson Edexcel International Advanced Subsidiary and Advanced Level examinations Pure Mathematics P1 - P4 Further Pure Mathematics FP1 - FP3 Mechanics M1 - M3
IBDP MATHEMATICS: ANALYSIS AND APPROACHES SYLLABUS SL 1.1 11 General SL 1.2 11 Mathematics SL 1.3 11 Mathematics SL 1.4 11 General 11 Mathematics 12 General SL 1.5 11 Mathematics SL 1.6 11 Mathematic12 Specialist SL 1.7 11 Mathematic* Not change of base SL 1.8 11 Mathematics SL 1.9 11 Mathematics AHL 1.10 11 Mathematic* only partially AHL 1.11 Not covered AHL 1.12 11 Mathematics AHL 1.13 12 .
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