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: (44) 01235 827720. Fax: (44) 01235 400454. 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