Pure Mathematics 2 and 3: Cambridge International AS and A Level Mathematics

353 Pages·2012·26.54 MB·English

by  Sophie Goldie

Cambridge
International AS and A Level Mathematics
Pure Mathematics 2 and 3
Sophie Goldie
Series Editor: Roger Porkess
Contents
Questions from the Cambridge International AS and A Level Mathematics papers
are reproduced by permission of University of Cambridge International Examinations.
Questions from the MEI AS and A Level Mathematics papers are reproduced by permission of OCR.
We are grateful to the following companies, institutions and individuals who have given permission
to reproduce photographs in this book.
Photo credits: page 2 © Tony Waltham / Robert Harding / Rex Features; page 51 left © Mariusz Blach – Fotolia;
page 51 right © viappy – Fotolia; page 62 © Phil Cole/ALLSPORT/Getty Images; page 74 © Imagestate Media Key to symbols in this book vi
(John Foxx); page 104 © Ray Woodbridge / Alamy; page 154 © VIJAY MATHUR/X01849/Reuters/Corbis;
Introduction vii
page 208 © Krzysztof Szpil – Fotolia; page 247 © erikdegraaf – Fotolia
The Cambridge International AS and A Level Mathematics syllabus viii
All designated trademarks and brands are protected by their respective trademarks.
Every effort has been made to trace and acknowledge ownership of copyright. The publishers
will be glad to make suitable arrangements with any copyright holders whom it has not been P2 Pure Mathematics 2 1
possible to contact.
Chapter 1 Algebra 2
®IGCSE is the registered trademark of University of Cambridge International Examinations.
Operations with polynomials 3
Hachette UK’s policy is to use papers that are natural, renewable and recyclable products and
made from wood grown in sustainable forests. The logging and manufacturing processes are Solution of polynomial equations 8
expected to conform to the environmental regulations of the country of origin.
The modulus function 17
Orders: please contact Bookpoint Ltd, 130 Milton Park, Abingdon, Oxon OX14 4SB.
Telephone: [phone]. Fax: [phone]. Lines are open 9.00–5.00, Monday
to Saturday, with a 24-hour message answering service. Visit our website at Chapter 2 Logarithms and exponentials 23
www.hoddereducation.com
Logarithms 23
Much of the material in this book was published originally as part of the MEI Structured Exponential functions 28
Mathematics series. It has been carefully adapted for the Cambridge International AS and A
Modelling curves 30
Level Mathematics syllabus.
The natural logarithm function 39
The original MEI author team for Pure Mathematics comprised Catherine Berry, Bob Francis,
Val Hanrahan, Terry Heard, David Martin, Jean Matthews, Bernard Murphy, Roger Porkess The exponential function 43
and Peter Secker.
Copyright in this format © Roger Porkess and Sophie Goldie, 2012
Chapter 3 Trigonometry 51
First published in 2012 by Reciprocal trigonometrical functions 52
Hodder Education, an Hachette UK company,
Compound-angle formulae 55
338 Euston Road
London NW1 3BH Double-angle formulae 61
Impression number 5 4 3 2 1 The forms r cos(θ ± α), r sin(θ ± α) 66
Year 2016 2015 2014 2013 2012
The general solutions of trigonometrical equations 75
All rights reserved. Apart from any use permitted under UK copyright law, no part of this
publication may be reproduced or transmitted in any form or by any means, electronic or
mechanical, including photocopying and recording, or held within any information storage Chapter 4 Differentiation 78
and retrieval system, without permission in writing from the publisher or under licence from The product rule 78
the Copyright Licensing Agency Limited. Further details of such licences (for reprographic
The quotient rule 80
reproduction) may be obtained from the Copyright Licensing Agency Limited, Saffron House,
6–10 Kirby Street, London EC1N 8TS. Differentiating natural logarithms and exponentials 85
Cover photo © Irochka – Fotolia Differentiating trigonometrical functions 92
Illustrations by Pantek Media, Maidstone, Kent Differentiating functions defined implicitly 97
Typeset in Minion by Pantek Media, Maidstone, Kent
Parametric equations 104
Printed in Dubai
Parametric differentiation 108
A catalogue record for this title is available from the British Library
ISBN 978 1444 14646 2
This eBook does not include the ancillary media that was packaged with the printed version of the book.
Contents
Key to symbols in this book vi
Introduction vii
The Cambridge International AS and A Level Mathematics syllabus viii
P2 Pure Mathematics 2 1
Chapter 1 Algebra 2
Operations with polynomials 3
Solution of polynomial equations 8
The modulus function 17
Chapter 2 Logarithms and exponentials 23
Logarithms 23
Exponential functions 28
Modelling curves 30
The natural logarithm function 39
The exponential function 43
Chapter 3 Trigonometry 51
Reciprocal trigonometrical functions 52
Compound-angle formulae 55
Double-angle formulae 61
The forms r cos(θ ± α), r sin(θ ± α) 66
The general solutions of trigonometrical equations 75
Chapter 4 Differentiation 78
The product rule 78
The quotient rule 80
Differentiating natural logarithms and exponentials 85
Differentiating trigonometrical functions 92
Differentiating functions defined implicitly 97
Parametric equations 104
Parametric differentiation 108
iii
Chapter 5 Integration 117 Chapter 11 Complex numbers 271
Integrals involving the exponential function 117 The growth of the number system 271
Integrals involving the natural logarithm function 117 Working with complex numbers 273
Integrals involving trigonometrical functions 124 Representing complex numbers geometrically 281
Numerical integration 128 Sets of points in an Argand diagram 284
The modulus–argument form of complex numbers 287
Chapter 6 Numerical solution of equations 136 Sets of points using the polar form 293
Interval estimation – change-of-sign methods 137 Working with complex numbers in polar form 296
Fixed-point iteration 142 Complex exponents 299
Complex numbers and equations 302
P3 Pure Mathematics 3 153
Answers 309
Index 341
Chapter 7 Further algebra 154
The general binomial expansion 155
Review of algebraic fractions 164
Partial fractions 166
Using partial fractions with the binomial expansion 173
Chapter 8 Further integration 177
Integration by substitution 178
Integrals involving exponentials and natural logarithms 183
Integrals involving trigonometrical functions 187
The use of partial fractions in integration 190
Integration by parts 194
General integration 204
Chapter 9 Differential equations 208
Forming differential equations from rates of change 209
Solving differential equations 214
Chapter 10 Vectors 227
The vector equation of a line 227
The intersection of two lines 234
The angle between two lines 240
The perpendicular distance from a point to a line 244
The vector equation of a plane 247
The intersection of a line and a plane 252
The distance of a point from a plane 254
The angle between a line and a plane 256
The intersection of two planes 262
iv
Chapter 11 Complex numbers 271
The growth of the number system 271
Working with complex numbers 273
Representing complex numbers geometrically 281
Sets of points in an Argand diagram 284
The modulus–argument form of complex numbers 287
Sets of points using the polar form 293
Working with complex numbers in polar form 296
Complex exponents 299
Complex numbers and equations 302
Answers 309
Index 341
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 symbol invites you to join in a discussion about proof. The answers to these
questions are given in the back of the book.
! 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 could use a graphic calculator or a
computer. Graphic calculators and computers are not permitted in any of the
examinations for the Cambridge International AS and A Level Mathematics 9709
syllabus, however, so these activities are optional.
This symbol and a dotted line down the right-hand side of the page indicate
material that you are likely to have met before. You need to be familiar with the
material before you move on to develop it further.
This symbol and a dotted line down the right-hand side of the page indicate
material which is beyond the syllabus for the unit but which is included for
completeness.
vi
Introduction
This is part of a series of books for the University of Cambridge International
Examinations syllabus for Cambridge International AS and A Level Mathematics
9709. It follows on from Pure Mathematics 1 and completes the pure mathematics
required for AS and A level. The series also contains a book for each of mechanics
and statistics.
These books are based on the highly successful series for the Mathematics in
Education and Industry (MEI) syllabus in the UK but they have been redesigned
for Cambridge international students; where appropriate, new material has been
written and the exercises contain many past Cambridge examination questions.
An overview of the units making up the Cambridge international syllabus is given
in the diagram on the next page.
Throughout the series the emphasis is on understanding the mathematics
as well as routine calculations. The various exercises provide plenty of scope
for practising basic techniques; they also contain many typical examination
questions.
An important feature of this series is the electronic support. There is an
accompanying disc containing two types of Personal Tutor presentation:
examination-style questions, in which the solutions are written out, step by step,
with an accompanying verbal explanation, and test-yourself questions; these are
multiple-choice with explanations of the mistakes that lead to the wrong answers
as well as full solutions for the correct ones. In addition, extensive online support
is available via the MEI website, www.mei.org.uk.
The books are written on the assumption that students have covered and
understood the work in the Cambridge IGCSE® syllabus. However, some
of the early material is designed to provide an overlap and this is designated
‘Background’. There are also places where the books show how the ideas can be
taken further or where fundamental underpinning work is explored and such
work is marked as ‘Extension’.
The original MEI author team would like to thank Sophie Goldie who has carried
out the extensive task of presenting their work in a suitable form for Cambridge
international students and for her many original contributions. They would
also like to thank University of Cambridge International Examinations for their
detailed advice in preparing the books and for permission to use many past
examination questions.
Roger Porkess
Series Editor
vii
The Cambridge International AS

and A Level Mathematics syllabus

P2
Cambridge
AS Level
IGCSE P1 S1
Mathematics
Mathematics
M1
M1
S1
S2
A Level
P3
Mathematics
S1
M1
M2
viii
Pure Mathematics 2
P2