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PEARSON EDEXCEL INTERNATIONAL A LEVEL

PURE MATHEMATICS 4

Student Book

Series Editors: Joe Skrakowski and Harry Smith

Authors: Greg Attwood, Jack Barraclough, Ian Bettison, Lee Cope, Charles Garnet Cox, Keith Gallick, Daniel Goldberg, Alistair Macpherson, Anne McAteer, Lee McKelvey, Bronwen Moran, Su Nicholson, Diane Oliver, Laurence Pateman, Joe Petran, Keith Pledger, Cong San, Joe Skrakowski, Harry Smith, Geoff Staley, Robert Ward-Penny, Dave WilkinsSAMPLE COPY Published by Pearson Education Limited, 80 Strand, London, WC2R 0RL. www.pearsonglobalschools.com Copies of ocial specications for all Pearson qualications may be found on the website: https://qualications.pearson.com

Text © Pearson Education Limited 2019

Edited by Linnet Bruce

Typeset by Tech-Set Ltd, Gateshead, UK

Original illustrations © Pearson Education Limited 2019

Illustrated by © Tech-Set Ltd, Gateshead, UK

Cover design by © Pearson Education Limited 2019 The rights of Greg Attwood, Jack Barraclough, Ian Bettison, Lee Cope, Charles Garnet Cox, Keith Gallick, DanielGoldberg, Alistair Macpherson, Anne McAteer, Lee McKelvey, Bronwen Moran, SuNicholson, Diane Oliver, Laurence Pateman, Joe Petran, Keith Pledger, Cong San, Joe Skrakowski, Harry Smith, Geo Staley, Robert Ward-Penny and Dave Wilkins to be identied as the authors of this work have been asserted by them in accordance with the

Copyright, Designs and Patents Act 1988.

First published 2019

22 21 20 19

10 9 8 7 6 5 4 3 2 1

British Library Cataloguing in Publication Data

A catalogue record for this book is available from the British Library

ISBN 978 1 292245 12 6

Copyright notice

All 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 and whether or not transiently or incidentally to some other use of this publication) without the written permission of the copyright owner, except in accordance with the provisions of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency, Barnard's Inn, 86 Fetter Lane, London, EC4A 1EN (www.cla.co.uk). Applications for the copyright owner"s written permission should be addressed to the publisher.

Printed by Neograa in Slovakia

Picture Credits

The authors and publisher would like to thank the following individuals and organisations for permission to reproduce photographs:

Alamy Stock Photo:

Terry Oakley 16;

Getty Images:

mikedabell 50, Westend61 97;

Science Photo Library:

Millard H. Sharp 66;

Shutterstock.com:

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LDprod 1, OliverSved 30

Cover images:

Front

Getty Images:

Werner Van Steen

Inside front cover

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All other images © Pearson Education Limited 2019

All artwork © Pearson Education Limited 2019Endorsement StatementIn order to ensure that this resource oers high-quality support for the associated Pearson qualication, it has been through a review process by the awarding body. This process conrms that this resource fully covers the teaching and learning content of the specication or part of a specication at which it is aimed. It also conrms that it demonstrates an appropriate balance between the development of subject skills, knowledge and understanding, in addition to preparation for assessment.

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 the teaching or delivery of a related course. While the publishers have made every attempt to ensure that advice on the qualication and its assessment is accurate, the ocial specication and associated assessment guidance materials are the only authoritative source of information and should always be referred to for denitive guidance. Pearson examiners have not contributed to any sections in this resource relevant to examination papers for which they have responsibility. Examiners will not use endorsed resources as a source of material for any assessment set by Pearson. Endorsement of a resource does not mean that the resource is required to achieve this Pearson qualication, nor does it mean that it is the only suitable material available to support the qualication, and any resource lists produced by the awarding body shall include this and other appropriate resources.SAMPLE COPY iiiCONTENTS

COURSE STRUCTURE iv

ABOUT

THIS BOOK

vi

QUALIFICA

TION AND ASSESSMENT OVERVIEW

viii

EXTRA ONLINE CONTENT

x 1 PROOF 1 2 P

ARTIAL FRACTIONS

6 3

COORDINA

TE GEOMETRY IN THE (

x y ) PLANE 16 4

BINOMIAL EXP

ANSION

30

REVIEW EXER

CISE 1

46
5

DIFFERENTIATION

50
6

INTEGRATION

66
7

VECTORS

97

REVIEW EXER

CISE 2

148

EXAM PRAC

TICE 153

GLOSSAR

Y 155

ANSWERS

159
INDEX

179SAMPLE COPY

ivCOURSE STRUCTURE

CHAPTER 1 PROOF 1

1.1 PROOF BY CONTRADICTION 2

CHAPTER REVIEW 1

5

CHAPTER 2 PARTIAL

FRACTIONS

6

2.1 PAR TIAL FRACTIONS 7

2.2 REPEA

TED FACTORS

10

2.3 IMPROPER FRA

CTIONS

12

CHAPTER REVIEW 2

14

CHAPTER 3 COORDINATE

GEOMETRY IN THE (

x y PLANE 16

3.1 PARAMETRIC EQUATIONS 17

3.2

USING TRIGONOMETRIC

IDENTITIES

21

3.3 CURVE SKET

CHING 25

CHAPTER REVIEW 3

28

CHAPTER 4 BINOMIAL

EXPANSION

30

4.1 EXPANDING (1 +

x n 31
4.2

EXPANDING (

a b x) n 36
4.3

USING PARTIAL FRACTIONS

40

CHAPTER REVIEW 4

43

REVIEW EXERCISE 1 46

CHAPTER 5 DIFFERENTIA

TION 50

5.1 PARAMETRIC DIFFERENTIATION 51

5.2 IMPLICIT DIFFERENTIA

TION 54

5.3 RA

TES OF CHANGE

57

CHAPTER REVIEW 5

61

CHAPTER 6 INTEGRATION 66

6.1 FINDING THE AREA UNDER A CURVE

DEFINED P

ARAMETRICALLY

67
6.2 V

OLUMES OF REVOLUTION AROUND

THE x-AXIS 68 6.3

INTEGRA

TION BY SUBSTITUTION

74

6.4 INTEGRA

TION BY PARTS

78
6.5 P

ARTIAL FRACTIONS

81
6.6 SOL

VING DIFFERENTIAL

EQUATIONS

84
6.7

MODELLING WITH DIFFERENTIAL

EQU

ATIONS

88

CHAPTER REVIEW 6

92SAMPLE COPY

vCOURSE STRUCTURE

CHAPTER 7 VECTORS 97

7.1 VECTORS 98

7.2 REPRESENTING VEC

TORS 102
7.3 MA

GNITUDE AND DIRECTION

106
7.4 VEC

TORS IN 3D

109
7.5 SOL

VING GEOMETRIC PROBLEMS

IN TWO DIMENSIONS

114
7.6 SOL

VING GEOMETRIC PROBLEMS

IN THREE DIMENSIONS

117

7.7 POSITION VEC

TORS 121

7.8 3D COORDINA

TES 123
7.9 EQUquotesdbs_dbs14.pdfusesText_20