PEARSON EDEXCEL INTERNATIONAL A LEVEL FURTHER PURE .
Student BookPLEFURTHER PUREMATHEMATICS 1Series Editors: Joe Skrakowski and Harry SmithAuthors: Greg Attwood, Jack Barraclough, Ian Bettison, Lee Cope, Charles Garnet Cox,Keith Gallick, Daniel Goldberg, Alistair Macpherson, Anne McAteer, Bronwen Moran,Su Nicholson, Laurence Pateman, Joe Petran, Keith Pledger, Cong San, Joe Skrakowski,Harry Smith, Geoff Staley, Dave WilkinsSAMUncorrected proof, all content subject to change at publisher discretion. Not for resale, circulation or distribution in whole or in part. Pearson 2019PEARSON EDEXCEL INTERNATIONAL A LEVELF01 IASL FPM1 44648 PRE i-x.indd 111/10/2018 16:06
Copies of official specifications for all Pearson qualifications may be found on thewebsite: https://qualifications.pearson.comText Pearson Education Limited 2018Designed by Pearson Education Limited 2018Typeset by Tech-Set Ltd, Gateshead, UKEdited by Eric PradelOriginal illustrations Pearson Education Limited 2018Illustrated by Tech-Set Ltd, Gateshead, UKCover design Pearson Education Limited 2018Endorsement 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.British Library Cataloguing in Publication DataA catalogue record for this book is available from the British LibraryISBN 978 1 292244 64 8Pearson 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.PLFirst published 2018While 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.EThe rights of Greg Attwood, Jack Barraclough, Ian Bettison, Lee Cope, CharlesGarnet Cox, Keith Gallick, Daniel Goldberg, Alistair Macpherson, Anne McAteer,Bronwen Moran, Su Nicholson, Laurence Pateman, Joe Petran, Keith Pledger,Cong San, Joe Skrakowski, Harry Smith, Geoff Staley and Dave Wilkins to beidentified as the authors of this work have been asserted by them in accordancewith the Copyright, Designs and Patents Act 1988.21 20 19 1810 9 8 7 6 5 4 3 2 1Endorsement 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.Copyright 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, Barnard's Inn, 86 FetterLane, London, EC4A 1EN (www.cla.co.uk). Applications for the copyright owner’swritten permission should be addressed to the publisher.SAMUncorrected proof, all content subject to change at publisher discretion. Not for resale, circulation or distribution in whole or in part. Pearson 2019Published by Pearson Education Limited, 80 Strand, London, WC2R 0RL.www.pearsonglobalschools.comPrinted in Slovakia by NeografiaPicture CreditsThe authors and publisher would like to thank the following individuals andorganisations for permission to reproduce photographs:Alamy Stock Photo: Paul Fleet 92; Getty Images: Anthony Bradshaw 36,David Trood 1, Dulyanut Swdp 49, gmutlu 116, jamielawton 76, Martin Barraud 127;Paul Nylander: 28Cover images: Front: Getty Images: Werner Van SteenInside front cover: Shutterstock.com: Dmitry LobanovAll other images Pearson Education Limited 2018All artwork Pearson Education Limited 2018F01 IASL FPM1 44648 PRE i-x.indd 211/10/2018 16:06
iiiCOURSE STRUCTURE ivABOUT THIS BOOK viEXTRA ONLINE CONTENT 1 COMPLEX NUMBERS viiixEQUALIFICATION AND ASSESSMENT OVERVIEW PL2 ROOTS OF QUADRATIC EQUATIONS 1283 NUMERICAL SOLUTIONS OF EQUATIONS 364 COORDINATE SYSTEMS 49REVIEW EXERCISE 1 715 MATRICES 766 TRANSFORMATIONS USING MATRICES 92SAMUncorrected proof, all content subject to change at publisher discretion. Not for resale, circulation or distribution in whole or in part. Pearson 2019CONTENTS7 SERIES 1168 PROOF 127REVIEW EXERCISE 2 141EXAM PRACTICE 145GLOSSARY 147ANSWERS 150INDEX 173F01 IASL FPM1 44648 PRE i-x.indd 311/10/2018 16:06
COURSE STRUCTURE12579111516494.1 PARAMETRIC EQUATIONS 504.2 THE GENERAL EQUATIONOF A PARABOLA 534.3 THE EQUATION FOR A RECTANGULARHYPERBOLA. THE EQUATION OF THETANGENT AND THE EQUATION OFTHE NORMAL 60CHAPTER REVIEW 4 68PL1.1 IMAGINARY AND COMPLEXNUMBERS 1.2 MULTIPLYING COMPLEX NUMBERS 1.3 COMPLEX CONJUGATION 1.4 ARGAND DIAGRAMS 1.5 MODULUS AND ARGUMENT 1.6 MODULUS-ARGUMENT FORMOF COMPLEX NUMBERS 1.7 ROOTS OF QUADRATIC EQUATIONS 1.8 SOLVING CUBIC ANDQUARTIC EQUATIONS CHAPTER REVIEW 1 CHAPTER 4 COORDINATESYSTEMS ECHAPTER 1 COMPLEXNUMBERS CHAPTER 2 ROOTS OFQUADRATIC EQUATIONS 2.1 ROOTS OF A QUADRATICEQUATION 2.2 FORMING QUADRATICEQUATIONS WITH NEW ROOTS CHAPTER REVIEW 2 18222829REVIEW EXERCISE 1 71CHAPTER 5 MATRICES 765.1 INTRODUCTION TO MATRICES 5.2 MATRIX MULTIPLICATION 5.3 DETERMINANTS 5.4 INVERTING A 2 2 MATRIX CHAPTER REVIEW 5 SAMUncorrected proof, all content subject to change at publisher discretion. Not for resale, circulation or distribution in whole or in part. Pearson 2019ivCHAPTER 3 NUMERICALSOLUTIONS OF EQUATIONS 3.1 LOCATING ROOTS 3.2 INTERVAL BISECTION 3.3 LINEAR INTERPOLATION 3.4 THE NEWTON-RAPHSON METHOD CHAPTER REVIEW 3 F01 IASL FPM1 44648 PRE i-x.indd 431343637394144477780858789CHAPTER 6 TRANSFORMATIONSUSING MATRICES 926.1 LINEAR TRANSFORMATIONSIN TWO DIMENSIONS 936.2 REFLECTIONS AND ROTATIONS 976.3 ENLARGEMENTS AND STRETCHES 1026.4 SUCCESSIVE TRANSFORMATIONS 1066.5 THE INVERSE OF A LINEARTRANSFORMATION 110CHAPTER REVIEW 6 11311/10/2018 16:06
CHAPTER 7 SERIES v1167.1 SUMS OF NATURAL NUMBERS 1177.2 SUMS OF SQUARES AND CUBES 120CHAPTER REVIEW 7 124127PL8.1 PROOF BY MATHEMATICALINDUCTION 1288.2 PROVING DIVISIBILITY RESULTS 1328.3 USING MATHEMATICAL INDUCTIONTO PRODUCE A PROOF FOR AGENERAL TERM OF ARECURRENCE RELATION 1348.4 PROVING STATEMENTSINVOLVING MATRICES 137CHAPTER REVIEW 8 139ECHAPTER 8 PROOF REVIEW EXERCISE 2 141SAMUncorrected proof, all content subject to change at publisher discretion. Not for resale, circulation or distribution in whole or in part. Pearson 2019COURSE STRUCTUREEXAM PRACTICE 145GLOSSARY 147ANSWERS 150INDEX 173F01 IASL FPM1 44648 PRE i-x.indd 511/10/2018 16:06
ABOUT 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 Notation boxes explain key mathematical language and symbolsThe Mathematical Problem-Solving Cycle2. Mathematical problem-solving Hundreds of problem-solving questions, fully integratedinto the main exercisesspecify the probleminterpret resultsPL Problem-solving boxes provide tips and strategies Challenge questions provide extra stretch3. Transferable skillsE Rigorous and consistent approach throughoutcollect informationprocess andrepresent information Transferable skills are embedded throughout this book, in the exercises and in some examples These skills are signposted to show students which skills they are using and developingFinding your way around the bookSAMUncorrected proof, all content subject to change at publisher discretion. Not for resale, circulation or distribution in whole or in part. Pearson 2019viEach chapter is mapped to thespecification content for easyreferenceEach chapter starts with alist of Learning objectivesThe Prior knowledgecheck helps make sureyou are ready to start thechapterThe real world applications of themathematics you are about tolearn are highlighted at the startof the chapterGlossary terms willbe identified by boldblue text on their firstappearanceF01 IASL FPM1 44648 PRE i-x.indd 611/10/2018 16:06
viiEach section beginswith explanation andkey learning pointsExam-style questionsare flagged with EProblem-solvingquestions are flaggedwith PPLStep-by-step workedexamples focus on thekey types of questionsyou’ll need to tackleETransferable skills aresignposted wherethey naturally occurin the exercises andexamplesExercise questions arecarefully graded so theyincrease in difficulty andgradually bring you upto exam standardExercises are packedwith exam-stylequestions to ensure youare ready for the examsEach chapter ends with a Chapter reviewand a Summary of key pointsSAMUncorrected proof, all content subject to change at publisher discretion. Not for resale, circulation or distribution in whole or in part. Pearson 2019ABOUT THIS BOOKAfter 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 thingF01 IASL FPM1 44648 PRE i-x.indd 711/10/2018 16:06
QUALIFICATION ANDASSESSMENT OVERVIEWQualification and content overviewEFurther Pure Mathematics 1 (FP1) is a compulsory unit in the following qualifications:International Advanced Subsidiary in Further MathematicsInternational Advanced Level in Further MathematicsAssessment overviewUnitPLThe 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.PercentageFP1: Further Pure Mathematics 1Paper code WFM01/01133 3 % of IASMarkTimeAvailability751 hour 30 minsJanuary and June216 3 % of IALFirst assessment June 2019IAS: International Advanced Subsidiary, IAL: International Advanced A Level.Assessment objectives and weightingsSAMUncorrected proof, all content subject to change at publisher discretion. Not for resale, circulation or distribution in whole or in part. Pearson 2019viii QUALIFICATION AND ASSESSMENT OVERVIEWMinimumweighting 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%F01 IASL FPM1 44648 PRE i-x.indd 811/10/2018 16:06
ixRelationship of assessment objectives to unitsAssessment objectiveFP1AO1AO2AO3AO4AO5Marks out of 7525–3025–300–55–10%33 3 –4033 3 –400–6 3 6 3 –13 3 11225–1016 3 –13 3 21CalculatorsEStudents 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.ProhibitionsPLStudents 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.Calculators 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 internetSAMUncorrected proof, all content subject to change at publisher discretion. Not for resale, circulation or distribution in whole or in part. Pearson 2019QUALIFICATION AND ASSESSMENT OVERVIEWF01 IASL FPM1 44648 PRE i-x.indd 911/10/2018 16:06
EXTRA ONLINE CONTENTExtra online contentWhenever you see an Online box, it means that there is extra online content available to support you.SolutionBankESolutionBank provides worked solutions for questions in the book.Download all the solutions as a PDF or quickly find the solution you need online.Use of technologyyxGeoGebra-powered interactivesgraphically using technology.PLExplore topics in more detail, visualiseproblems and consolidate your understanding.Use pre-made GeoGebra activities or Casioresources for a graphic calculator.Online Find the point of intersectionGraphic calculator interactivesInteract with the mathsyou are learning usingGeoGebra's easy-to-usetoolsSAMUncorrected proof, all content subject to change at publisher discretion. Not for resale, circulation or distribution in whole or in part. Pearson 2019xInteract with the mathematics you are learningusing GeoGebra's easy-to-use toolsExplore the mathematics you are learning andgain confidence 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.F01 IASL FPM1 44648 PRE i-x.indd 10Step-by-step guide with audio instructionson exactly which buttons to press and whatshould appear on your calculator's screen11/10/2018 16:06
CHAPTER 11Learning objectivesAfter completing this chapter you should be able to:Understand and use the definitions of imaginary and complex numbers page 2Add and subtract complex numbers pages 2–3Find solutions to any quadratic equation with real coefficients pages 4–5Multiply complex numbers pages 5–6Understand the definition of a complex conjugate pages 7–8Divide complex numbers pages 7–8Show complex numbers on an Argand diagram pages 9–10Find the modulus and argument of a complex number pages 11–14Write a complex number in modulus-argument form pages 15–16Solve quadratic equations that have complex roots pages 16–18Solve cubic or quartic equations that have complex roots pages 18–22PL Prior knowledge check1Simplify each of the following:a21.11.21.31.41.51.6E1 COMPLEXNUMBERSSAMUncorrected proof, all content subject to change at publisher discretion. Not for resale, circulation or distribution in whole or in part. Pearson 2019COMPLEX NUMBERS 50b 108c 180 Pure 1 Section 1.5In each case, determine the number of distinctreal roots of the equation f(x) 0.a f(x) 3x 2 8x 10b f(x) 2x 2 9x 7c f(x) 4x 2 12x 93For the triangle shown, find the values of:b θa x5 cmθx12 cm45 Pure 1 Section 2.3 International GCSE MathematicsFind the solutionsof 8x 6 0, giving your answers in theform a b where a and b are integers. Pure 1 Section 2.17Writein the form p q 34 3where p and q are rational numbers. Pure 1 Section 1.6M01 IASL FPM1 44648 U01 001-027.indd 1x2Complex numbers contain areal part and an imaginarypart. Engineers and physicistsoften describe quantities withtwo components using a singlecomplex number. This allowsthem to model complicatedsituations such as air flow overa cyclist.11/10/2018 16:16
CHAPTER 1COMPLEX NUMBERS1.1 Imaginary and complex numbersThe quadratic equation a x 2 bx c 0 hassolutions given by b b 2 4ac x For the equation a x 2 bx c 0 ,the discriminant is b 2 4ac . If b 2 4ac 0 , there are two distinct real roots. If b 2 4ac 0 , there are two equal real roots. If b 2 4ac 0 , there are no real roots. Pure 1 Section 2.5E2aIf the expression under the square root is negative,there are no real solutions.LinksExample1SKILLSPLYou can find solutions to the equation in all cases by extending the numbersystem to include 1 .Since there is no real number that squares to produce 1, the number 1 is called an imaginarynumber, and is represented using the letter i. Complex numbers have a real part and an imaginarypart, for example 3 2i.Notation The set of all complex i 1 numbers is written as ℂ. An imaginary number is a numberFor the complex number z a bi:of the form bi, where b ℝ. Re(z) a is the real part A complex number is written in Im(z) b is the imaginary partthe form a bi, where a, b ℝ.INTERPRETATIONWrite each of the following in terms of i.b 28 a 36 28 1 4 7 1 You can use the rules of surds to manipulateimaginary numbers. 36 a 36 ( 1) 36 1 6iSAMUncorrected proof, all content subject to change at publisher discretion. Not for resale, circulation or distribution in whole or in part. Pearson 20192 b 28 28 ( 1) Watch out ( 2 7 ) i An alternativeway of writing ( 2 7 ) i is 2i 7 . Avoid writing2 7 i asthiscaneasilybeconfused with 2 7i .In a complex number, the real part and the imaginary part cannot be combined to form a single term. Complex numbers can be added or subtracted by adding or subtractingtheir real parts and adding or subtracting their imaginary parts. You can multiply a real number by a complex number by multiplying outthe brackets in the usual way.Example2Simplify each of the following, giving your answers in the form a bi, where a, b ℝ .10 6ia (2 5i) (7 3i)b (2 5i) (5 11i)c 2(5 8i)d 2a (2 5i) (7 3i) (2 7) (5 3)i 9 8ib (2 5i) (5 11i) (2 5) ( 5 ( 11))i 3 6iM01 IASL FPM1 44648 U01 001-027.indd 2Add the real parts and add the imaginary parts.Subtract the real parts and subtract theimaginary parts.11/10/2018 16:16
CHAPTER 1c 2(5 8i) (2 5) (2 8)i 10 16i10 6i10 6d i 5 3i 222Exercise1ASKILLS32(5 8i) can also be written as (5 8i) (5 8i).First separate into real and imaginary parts.INTERPRETATIONDo not use your calculator in this exercise.a 9 b 49 c 121 f 5 g 12 h 45 E1 Write each of the following in the form bi, where b is a real number.d 10 000 e 225 i 200 j 147 2 Simplify, giving your answers in the form a bi , where a, b ℝ .c (7 6i) ( 3 5i)b (4 10i) (1 8i)PLa (5 2i) (8 9i)d ( 2 3 i) ( 2
Further Pure Mathematics 1 (FP1) is a compulsory unit in the following qualifications: International Advanced Subsidiary in Further Mathematics International Advanced Level in Further Mathematics Assessment overview The following table gives an overview of the assessment for this unit.
Edexcel International A Level Mathematics Pure 3 Student Book 978 1 292244 92 1 21.00 Edexcel International A Level Mathematics Pure 3 Teacher Resource Pack 978 1 292244 93 8 100.00 Edexcel International A Level Mathematics Pure 4 Student Book 978 1 292245 12 6 21.00 Edexcel International A Level Mathematics Pure 4 Teacher Resource Pack
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Pearson Edexcel Certificate Pearson Edexcel International GCSE Turn over . 2 *P48391A0228* DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Answer ALL questions.
