CambridgeInternational AS A LevelFurtherMathematicsFurther PureMathematics 2Jean-Paul MuscatRose JewellSeries editor:Roger PorkessDDYNAMICHODDEREDUCATIONAN HACHETTE UK COMPANYQuestions from the Cambridge International AS A Level Further Mathematics papers are reproducedby permission of Cambridge Assessment International Education.Unless otherwise acknowledged,thequestions,example answers,and comments that appear in this book were written by the authors.Cambridge Assessment International Education bears no responsibility for the example answers toquestions taken from its past question papers which are contained in this publication.The Publishers would like to thank the following for permission to reproduce copyright material.Photo creditsp.1 imatthiasschulz-iStock via Thinkstock:p.16 Alexander Zelnitskiy/123RF P.60 tl,tr,bl Geograph.com br David Burrows Shutterstock p.85 Digoarpi-iStock via Thinkstock;p.117 Olaf Schulz/stock.adobe.com;p.145 World History Archive/Alamy Stock Photo.t-top,b bottom,(left,r right.Every effort has been made to trace and acknowledge ownership of copyright.The publishers will beglad to make suitable arrangements with any copyright holders whom it has not been possible tocontact.Although every effort has been made to ensure that website addresses are correct at time of going topress,Hodder Education cannot be held responsible for the content of any website mentioned in thisbook.It is sometimes possible to find a relocated web page by typing in the address of the home pagefor a website in the URL window of your browser.Hachette UK's policy is to use papers that are natural,renewable and recyclable products and madefrom wood grown in sustainable forests.The logging and manufacturing processes are expected toconform to the environmental regulations of the country of origin.Orders:please contact Bookpoint Ltd,130 Milton Park,Abingdon,Oxon 0X14 4SE.Telephone:(44)01235 827720.Fax:(44)01235 400401.Email education@bookpoint.co.uk Lines are open from9 a.m.to 5 p.m.,Monday to Saturday,with a 24-hour message answering service.You can also orderthrough our website:www.hoddereducation.com.Much of the material in this book was published originally as part of the MEI Structured Mathematicsseries.It has been carefully adapted for the Cambridge International AS A Level Further Mathematicssyllabus.The original MEI author team for Pure Mathematics comprised Catherine Berry,Val Hanrahan,Terry Heard,David Martin,Jean Matthews,Roger Porkess and Peter Secker.Roger Porkess,Jean-Paul Muscat and Rose Jewell 2018First published 2018 byHodder Education,An Hachette UK CompanyCarmelite House50 Victoria EmbankmentLondon EC4Y ODZwww.hoddereducation.comImpression number 10 9 8 7 6 5 4 3 2 1Year20222021202020192018All rights reserved.Apart from any use permitted under UK copyright law,no part of this publicationmay be reproduced or transmitted in any form or by any means,electronic or mechanical,includingphotocopying and recording,or held within any information storage and retrieval system,withoutpermission in writing from the publisher or under licence from the Copyright Licensing Agency Limited.Further details of such licences(for reprographic reproduction)may be obtained from the CopyrightLicensing Agency Limited,www.cla.co.ukCover photo Comaniciu Dan/ShutterstockIllustrations by Aptara,Inc.and Integra Software ServicesTypeset in Bembo std 11/13 Integra Software Services Pvt Ltd,Pondicherry,IndiaPrinted in ItalyA catalogue record for this title is available from the British Library.I5BN:9781510421790MIXPaper fromresponsible sourcesFSCwww.fsc.orgFSC"C104740ContentsIntroductionHow to use this bookviThe Cambridge International AS A Level FurtherMathematics 9231 syllabus1 Hyperbolic functions11.1Hyperbolic functions11.2Inverse hyperbolic functions82 Matrices162.1Important results relating to 2 x 2 matrices162.2Finding the inverse of a 3 x 3 matrix232.3Intersection of three planes292.4Eigenvalues and eigenvectors383 Differentiation603.1Differentiating inverse trigonometric functions613.2Differentiating hyperbolic functions623.3Differentiating inverse hyperbolic functions623.4Finding the second derivative for relations given implicitly643.5Finding the second derivative for relations given parametrically 673.6Polynomial approximations and Maclaurin series703.7Using the Maclaurin series for standard functions794 Integration854.1Integration using the inverse sine function854.2Integrating hyperbolic functions884.3Integration using inverse hyperbolic functions894.4Integration using reduction formulae924.5Approximation of areas using rectangles974.6Using areas under curves to approximate series sums994.7The arc length of a curve1024.8 Surface area of a solid of revolution109