A and AS Math.pdf

-. _._, / ,- ' ' . : '. ··., Advanced level Mathematics PUre Mclthematics .l .· i~~~,- \,0 ·-, HCAMBRIDGE V UNIVERSITY PRESS ~
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   . ' . : ' · dvanced l v l athematics PUre clthematics l · HC MBRIDGE V UNIV RSITY PR SS i~~~ \,0 _._ ·-  · I · L i r i r· i · ......--   ~ : · q CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo New Delhi Cambridge University Press The Edinburgh Building, Cambridge CB2 2RU, UK Information on this title: © Cambridge University Press 2002 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no ·reproduction of any part may take place without the written permission of Cambridge University Press. First published 2002 5th printing 2005 First South Asian edition 2006 Reprinted 2007 (twice), 2008 (twice), 2009 (twice) Printe4 in India by Replika Press Pvt. Ltd. A catalogue record or this publication s available from the British Library ISBN-13: 978 0 521 69637 l paperback This edition is for sale in South and South East Asia only, not for export elsewhere. Cover image: ©James L Amos/CORBIS  Contents Introduction v Coordinates points and lines l Surds and indices . Functions and graphs \v4 uadratic~ v Inequalities Revision exercise 1 · · Differentiation 7 ~pplications of differentiation vS/ Sequences · · The binomial theorem . O Trigonometry Combining and inverting functions 12 Extending differentiation Revision exercise 2 \} _,/ Vectors 14 Geometric sequences 5 Second derivatives 6 Integration 7 Volume of revolution 18 Radians Revision exercise 3 Practice examinations Answers Index b ~ 1 · c I c~ 1 - iv 1 7 32 Si 65 73 75 95 4 28 38 156 174 187 190 210 225 236 258 264 279 283 288 313 /  \ ntroduction Cambridge International Examinations CIE) Advanced Level Mathematics has been created especially for the new CIE mathematics syllabus. There is one book corresponding to each syllabus unit, exceptthat units P2 and P3 are contained in a single book. This book covers the first Pure Mathematics unit, Pl. The syllabus content is arranged by chapters which are ordered so as to provide a viable teaching course. The early chapters develop the foundations of the syllabus; students may already be familiar with some of these topics. Later chapters, however, are. largely independent .of each other, and teachers may wish to vary the order in wh~ h they are used. Some chapters, particularly Chapters 2, 3 and the first four sections of Chapter 8, contain material which is not in the examination syllabus for Pl and which therefore cannot be the direct focus of examination questions. Some of this is necessary background material, such as indices and surds; some is useful knowledge, such as graphs of powers of x, the use and meaning of modulus, and work on sequences. A few sections include important results which are difficult to prove or outside the syllabus. These sections are marked with an asterisk(*) in the section heading, and there is usually a sentence early on explaining precisely what it is that the student needs to know. Occasionally within the text paragraphs appear in this type style These paragraphs are usually outside the main stream of the mathematical argument, but may help to give insight, or suggest extra work or different approaches. Graphic calculators are not permitted in the examination, but they are useful aids in learning mathematics. In the book the authors have noted where access to a graphic calculator would be especially helpful but have not assumed that they are available to all students. Numerical work is presented in a form intended to discourage premature approximation. In· ongoing calculations inexact numbers appear in decimal form like 3.456 . . signifying that the number is held in a calculator to more places than are given. Numbers are not rounded at this stage; the full display could be, for example, 1.456123 or 3 .456 789. Final answers are then stated with some indication that they are approximate, for example 1.23 correct to 3. significant figures . There are plenty of exercises, and each chapter ends with a Miscellaneous exercise which includes some questions of examination standard. Three Revision exercises consoliate work in preceeding chpaters. The book concludes with two Practice examination papers. In some exercises a few of the later questions may go beyond the likely requirements of the Pl examination, either in difficulty or in length or both. Some questions are marked with an < tSterisf which indicates that they require knowledge ofresults outside the syllabus. Cambridge U_niversity Press would like to thank OCR (Oxford, Cambridge and RSA Examinations), part of the University of Cambridge Local Examinations Syndicate (UCLES) group, for permission to use past examination questions set in the United Kingdom. The authors thank UCLES and Cambridge University Press, in particular Diana Gillooly, for their help in producing this book. However, the responsibility for the text, and for any errors, remains with the authors.
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