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Cambridge IGCSE Mathematics Core and Extended PDF

560 Pages·2018·41.584 MB·English
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Dynamic Learning is an online subscription solution that supports teachers and students with high quality content and unique tools. Dynamic Learning incorporates elements that all work together to give you the ultimate classroom and homework resource. Online Teacher’s Guides include a host of informative and practical teaching resources, such as: ● Lesson planning support via editable schemes of work ● Teaching materials, for example worksheets or glossaries ● Answers, extra teaching notes and/or exam-style questions Cambridge IGCSE® Core and Extended Mathematics Fourth edition is available as a Whiteboard eTextbook which is an online interactive version of the printed textbook that enables teachers to: ● Display interactive pages to their class ● Add notes and highlight areas ● Add double-page spreads into lesson plans Additionally the Student eTextbook of Cambridge IGCSE® Core and Extended Mathematics Fourth edition is a downloadable version of the printed textbook that teachers can assign to students so they can: ● Download and view on any device or browser ● Add, edit and synchronise notes across two devices ● Access their personal copy on the move To find out more and sign up for free trials visit: www.hoddereducation.com/dynamiclearning IGCSE® Cambridge Mathematics Core and Extended Fourth edition Ric Pimentel Terry Wall 9781510421684.indb 1 22/02/18 1:54 PM ®IGCSE is a registered trademark The Publishers would like to thank the following for permission to reproduce copyright material. Photo credits p 2 © Aleksandra Antic/Shutterstock; p 3 © Dinodia Photos/Alamy Stock Photo; p 100 © katjen/Shutterstock; p 101 © Eduard Kim/Shutterstock; p 248 © Halfpoint/Shutterstock; p 249 © Georgios Kollidas/Fotolia; p 280 © ESB Professional/Shutterstock; p 281 © Classic Image/Alamy Stock Photo; p 344 © WitR/Shutterstock; p 345 © Hirarchivum Press/Alamy Stock Photo; p 390 © 3Dsculptor/Shutterstock; p 391 © Granger, NYC/TopFoto; p 438 © Anton Petrus/ Shutterstock; p 439 © Science History Images/Alamy Stock Photo; p 474 © Harvepino/Shutterstock; p 475 © Bernard 63/ Fotolia; p 502 © Shutterstock; p 503 © Jason Butcher/Getty Images Acknowledgements Every effort has been made to trace all copyright holders, but if any have been inadvertently overlooked, the Publishers will be pleased to make the necessary arrangements at the first opportunity. All exam-style questions and sample answers in this title were written by the authors. Although every effort has been made to ensure that website addresses are correct at time of going to press, Hodder Education cannot be held responsible for the content of any website mentioned in this book. It is sometimes possible to find a relocated web page by typing in the address of the home page for 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 made from wood grown in sustainable forests. The logging and manufacturing processes are expected to conform to the environmental regulations of the country of origin. Orders: please contact Bookpoint Ltd, 130 Park Drive, Milton Park, Abingdon, Oxon OX14 4SE. Telephone: (44) 01235 827720. Fax: (44) 01235 400401. Email [email protected] Lines are open from 9 a.m. to 5 p.m., Monday to Saturday, with a 24-hour message answering service. You can also order through our website: www.hoddereducation.com  Ric Pimentel and Terry Wall 1997, 2006, 2013, 2018 First published in 1997 Second edition published in 2006 Third edition published in 2013 This edition published in 2018 by Hodder Education, An Hachette UK Company Carmelite House 50 Victoria Embankment London EC4Y 0DZ www.hoddereducation.co.uk Impression number 10 9 8 7 6 5 4 3 2 1 Year 2022 2021 2020 2019 2018 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 and retrieval system, without permission 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 Copyright Licensing Agency Limited, www.cla.co.uk Cover photo © Shutterstock/ju.grozyan Illustrations by Pantek Media and Integra Software Services Typeset in Times Ten LT Std Roman 10/12 by Integra Software Servises Pvt. Ltd., Pondicherry, India Printed in Slovenia A catalogue record for this title is available from the British Library. ISBN: 978 1 5104 2168 4 9781510421684.indb 2 22/02/18 1:54 PM Contents Introduction v How to use this book v TOPIC 1 Number 2 Chapter 1 Number and language 4 Chapter 2 Accuracy 13 Chapter 3 Calculations and order 24 Chapter 4 Integers, fractions, decimals and percentages 30 Chapter 5 Further percentages 43 Chapter 6 Ratio and proportion 50 Chapter 7 Indices and standard form 62 Chapter 8 Money and finance 73 Chapter 9 Time 83 Chapter 10 Set notation and Venn diagrams 85 Topic 1 Mathematical investigations and ICT 96 TOPIC 2 Algebra and graphs 100 Chapter 11 Algebraic representation and manipulation 102 Chapter 12 Algebraic indices 122 Chapter 13 Equations and inequalities 126 Chapter 14 Linear programming 147 Chapter 15 Sequences 153 Chapter 16 Proportion 166 Chapter 17 Graphs in practical situations 172 Chapter 18 Graphs of functions 193 Chapter 19 Differentiation and the gradient function 219 Chapter 20 Functions 239 Topic 2 Mathematical investigations and ICT 245 TOPIC 3 Coordinate geometry 248 Chapter 21 Straight line graphs 250 Topic 3 Mathematical investigations and ICT 277 TOPIC 4 Geometry 280 Chapter 22 Geometrical vocabulary and construction 282 Chapter 23 Similarity and congruence 293 Chapter 24 Symmetry 310 Chapter 25 Angle properties 318 Topic 4 Mathematical investigations and ICT 341 TOPIC 5 Mensuration 344 Chapter 26 Measures 346 Chapter 27 Perimeter, area and volume 351 Topic 5 Mathematical investigations and ICT 386 9781510421684.indb 3 22/02/18 1:54 PM CONTENTS TOPIC 6 Trigonometry 390 Chapter 28 Bearings 392 Chapter 29 Trigonometry 394 Chapter 30 Further trigonometry 418 Topic 6 Mathematical investigations and ICT 436 TOPIC 7 Vectors and transformations 438 Chapter 31 Vectors 440 Chapter 32 Transformations 452 Topic 7 Mathematical investigations and ICT 471 TOPIC 8 Probability 474 Chapter 33 Probability 476 Chapter 34 Further probability 488 Topic 8 Mathematical investigations and ICT 498 TOPIC 9 Statistics 502 Chapter 35 Mean, median, mode and range 504 Chapter 36 Collecting, displaying and interpreting data 510 Chapter 37 Cumulative frequency 532 Topic 9 Mathematical investigations and ICT 541 Index 544 iv 9781510421684.indb 4 22/02/18 1:54 PM Introduction This book has been written for all students of Cambridge IGCSE® and IGCSE (9-1) Mathematics syllabuses (0580/0980). It carefully and precisely follows the syllabus from Cambridge Assessment International Education. It provides the detail and guidance that are needed to support you throughout your course and help you to prepare for your examinations. How to use this book To make your study of mathematics as rewarding and successful as possible, this Cambridge endorsed textbook offers the following important features: Learning objectives » Each topic starts with an outline of the subject material and syllabus objectives to be covered. Organisation » Topics follow the order of the syllabus and are divided into chapters. Within each chapter there is a blend of teaching, worked examples and exercises to help you build confidence and develop the skills and knowledge you need. At the end of each chapter there are comprehensive Student Assessments. You will also find short sets of informal, digital questions linked to the Student eTextbook, which offer practice in topic areas that students often find difficult. ICT, Mathematical modelling and problem-solving » The syllabus specifically refers to ‘Applying mathematical techniques to solve problems’, and this is fully integrated into the exercises and assessments in the book. There are also sections called ‘Mathematical Investigations and ICT’, which include problem-solving questions and ICT activities (although the latter would not be part of the examination). In the Student eTextbook there is a selection of videos which offer support in problem-solving strategies and encourage reflective practice. v 9781510421684.indb 5 22/02/18 1:54 PM Callouts Worked examples Exercise These commentaries The worked examples cover These appear throughout the text, provide additional important techniques and question and allow you to apply what you have explanations and styles. They are designed to learned. There are plenty of routine encourage full reinforce the explanations, and questions covering important understanding of give you step-by-step help for examination techniques. mathematical principles. solving problems. 11 AlgebrAic representAtion And mAnipulAtion 10 Set notation and Venn The universal set Exercise 11.11 Factorise the following quadratic expressions: diagrams However, sets C, D, E, F, G and H are considered proper subsets of A. 1 ad xx22 +− 77xx ++ 1122 eb xx22 +− 88xx ++ 1122 fc xx22 +− 1133xx ++ 1122 ThisC d ist iAnc atinodn Dof su Abs eett ci.s shown in the notation below. 2345 aaaadddd xxxxxxxx22222222 +++++−−− 61112x2x45x0x x −−xx x +−− 14+++ 51282 2352465 eeeebbbb xxxxxxxx22222222 ++++−+−− 621224x2x210xx x− xxx +−−− 2+++ 81106 5123232461 ffffcccc xxxxxxxx22222222 +−−−+−++ 6111x1x35x302 x−−x xxx +− 13++++ 9205 342346246 SAsarey s pmeseWretbte tSos sioe,ls snt ra htak eiwrsdee ei scald lasr- ledeelpee xdrfie na tsSehem.den tgeperldeol meuapses neo tfs oo Sbf. j teIhcf etes s doeort. e sIsyf mnaonbt o ebllsee. lmTohneeng tot oeb jbseeectlt oSsn otghrs i st ois S Aa i m=LW {ii1lsa,t o r2sl,ury 3b,k , s4ee,t GG5 dB, 6 {e,e 7 v,xHH e8an, ii9 mmmn, u1ppm0pll}iibleeeessr stthh}.aatt GG iiss nnoott aa spurobpseetr osfu Hbset of H 6 adg 224xxx222 +−+ 371xx2 x++ + 16 9 ehb 392xxx222 ++− 786xxx +++ 641 fic 362xxx222 ++− x1x 1 −−x 61− 4 a A i p{DaSreotsuicctruhilb Aaer f tsrheiceta cs, oeNtn.asmistisb ioaf, tEhgey fpotl,l Aownignogl ae,l e..m.}ents: b LiCBst ==su {{b22s,, e34t,, 56C,, 78{p}, r1i0m}e numbers}. ReWaorrrkaendg eexmamenptle osf complex formulae i iii i TAeIs.hg dte.hd Z e eai lmsneemobtt aehfibnenwtrist et eow, ofGo rt hh eineale nfismanetiet naert?se tcoo uthnet rsieets. of Africa. Exercise 10.2 1 Pab =LL {iiwsstth ttohhleee ssnuuubbmsseebtte QRrs {{loeedvssed nt nh nuaumnm 3b0be}ersr}s.}. Ma a2kCCπe ==th2reπ rletters in red the subject of eachb fo±rAmAπAπu==la=πr:r2r2 b Ci i i onDTWeF.sihgenirde.is it3 tceeee rra l.i debtTnhmoedhew eet 1hsnnr5eee ttt s wi s sAoe oaft .= etfih l{neexim :t sexee tnni satu sarm e on bfta hetthreu eorn afsa lect nuto.uruamnl tbnreiuerms} ibne Ars.frica. 2 cdeAabc =LLLLLL {iiiiiiwsssssstttttth ttttttohhhhhhleeeeeee ssssssnuuuuuuubbbbbbmsssssseeeeeebtttttte STUBCDr s { {{{ {{psbmmtsqrrqeiiuuutaumwllanatteriiegrppe eeln lle nneeun ss unmu5 muoom0bmff ba eb53ebnre}}sred..r}ssr .}7}s..}0.} NSqoutaer:e n obot tyh = s hid e−s x ec xm m2 2 m +xx 2 yy ====y223999===maaaax22±2h2hppp2xp2−h2x−2x2 df AAf 2f ( k=f2 p p === p+ y+ +x kqx+qqkq2xx222)===yyyAA+++xxx−p dc CCi i i i i i oonnD TlWeDTWei..ssnhhggeeiirreddee..iiss tt (5 ceecweeee0 rrrra ll,iii ddee bbtttn–hhhmmooeed4e eww )e eett 6 hhqssnnnna.ee3eeu ttnttttss awwd ss CBootee ooi(fftt o1 == ..eettn0 hhll{{ ,ee x(yee1mmx: 6=ss,2 ee)ee y 2ttnn) xaitt: n ssry–x ceoo l=4 ffut. h2dtt 8hhexe} ee c–a oss n4eeoy}ttr ..dniunmatbeesr o bfe ptowienetsn f2o uanndd 8o nin tchleu ssitvrea.ight 34T hJabSabcdefgh t e=a LL{{{{{{{{t{ Am11v4p1epuii,,o,2oss ,alw 226,ttltgq nn l,,,a1aahee ,g338 t3llyreriollo,,,,i} b vtae441tt1,h ahh,s}}0b4e e l,Mee}, a⊆lr c ,r 1n⊄ spa boe 5{uas{rr a1az}1{ronb ac,4sao,⊄ pa skh2m,2t e}e e,6 ls,{ to br}⊆3wt, 3s fbis,8 ⊆s, qho a4{tu,4 ufhoefl}1{b}r leevlJ0}uset } .e}eif tgn⊆ot}se ul { lt{otmoacefwoba bJulmiee.nnrs gsts} r}psiteoasrt tei}nm Aenfrtisc ais} true or false: q=±yA+x−p Tcohnet auinnisv earlls athl ese pt o(sEs )ib floer e alnemy penartsti cfourla trh patr opbrolebmle ims .the set which The complement of a set A is the set of elements which are in E but not in A. The complement of A is identified as A'. Notice that E' = ∅ and ∅' = E. 112 85 87 9781510421684_Ch11.indd 112 30/12/17 6:43 PM9781510421684_Ch10.indd 85 30/12/17 5:55 PM9781510421684_Ch10.indd 87 30/12/17 5:55 PM Material with a blue dotted line Student Mathematical is for material that is part of assessment both the Cambridge IGCSE Core investigations and ICT and Cambridge IGCSE Extended End-of-chapter questions More problem solving activities Curriculum. Material with a to test your understanding are provided at the end of each purple dotted line is for material section to put what you've of the key topics and help to that is only part of the Cambridge learned into practice. prepare you for your exam. IGCSE Extended curriculum. 10 Set notation and Venn diagramS 1 Mathematical Simplifying complex algebraic fractions Exercise 10.3 (cont) 3 Cquoemstpiolent e2 tahbeo svtea.tement A  B = {...} for each of the Venn diagrams in investigations and ICT Student assessment 4 4 1 Tish teh ev ohleuimghet Vof o tfh ae ccyylliinnddeerr iasn gdi vre ins tbhye t rhaed fiours.mula V = πr 2h, where h ab rps qtA ImaSdnnoiafvdmfite hctseuhettilmeigtmn aat eottiinis ock vwnanelshos dtaewiirngs ehc af otoaaevncwsee . i rdtmio ew psso itstarhtrte atam.n Tm tfh rpaeoat mhsrtter maounfca tmitudicareaetahl a etinnhmdvaae te tsaxitc iamgamala tlptieholaeenr mb,n ieiatnl tocgiacw. niAa mnslle a heyam s 2 abcT heFdMAc oiifa naorcmrdkyrem eletci hntture e dlttra oehv o refo3 of ol s rsu2fu. mf0tbh.h cjeeeemi cgos.thuf orta ff6 a c0tcyh eclei mna frd oherraaimc soa uafl l apva oo.cslluot ms6e.e5d mocyf ll5oin5nd0ge0 ar c nmisd 3A :w fi =int hd2 π airt s( rr a+d hiu),s help you. where r is the radius of the cylinder and h is its height. 5 Ca oEpy A= a {n..d7.}5 compl2e3te th14e fo8bll owABi'n =g {s.t..a}tements: 12345678 RDPLTETCreuoxherystapoae t twtdr ckothek h t fess fiheotsi mhrn errye ad uapoqst l ulu aupetel reh atg f dssetroet tiu fnrieaorol ergeoarnnrir mgan aa ciillmen ang srw reiasyumeb lolte feqropuuxa u lliarhielne cymre asc ewlatpalpsilnoslouy.eedr.nl.dts ss hsi.t.naa sra tbn we oeitrnhd easrnimesdwp telear becdlae.s.es. 3 abcTw hhdeF2RW5o= 0 sicfeneh omcad armdr?m xtrti , amh2 iGugsne+ lieitg avvshney iuefnes 2o rti gyohhfr+ oa nyeeficuzo sinfeg r2uoa dh aarrrit remnn aeo sgnxuafw ,s lt aowaeyh f r eceta yao trnlol ecitdm nnoy3 dzgal 3is tken.i hfsrsed..: fdoeh. rf o tsohfu fet rhr fsaaeudc bbeiujo easdc r1yte.2 ad c i5ma0g 0oa cnnmadl 2 h oaefni gadh crtua bdoiuids 6 Worked example ab FHionwd dlo wnhg eisn txh e= b2o, yd y= d3i aagnodn za l= o 4f .a block of concrete in the shape Cad oEApy = a {n .B.d.} =co {m...}plete the foebll ow(AAi'n =∩g { s.Bt..a)}t'e =m {e..n.}ts: cf AA ∩∩ BB '= = { {.....}.} Acoitr hmceuyrm sptfoiecir nreotn.s cTeeh ioes fcd ariea cagitrrecadlme .b (Sylt eprfaltai)gc shihnto glwi ans en sau ammreyb setthri ceo nrfo dpsroeai nwwtinst hefr v2oe0mn pl yeo aisncphtas .cpeodin otn t oth eev ery cd oRFfien aad rr rxea cwntaghnee gntuh dlea =rfo p0rr.m8is6um, lya o =tfo 0d m.i2ma5ek anens xdio tznh s=e 2 s0 mu.4b,1 j3.e mct. and 75 cm? 6 A 1412048 612 3915B abTa1n o/d HHa2 cnooos.wwwu Tn emmtrr aatthhnnyeeyy s nsssettu rriqmaamuiibggeehhsprttti oolliilnfnne seeli,ss n y ceawosruoa.e u atslhrdeee trnhesoe?tr: ee xbpee ocnte ad mtoy dstriacw r oesieth werit ohf 1 t0h0e pshoainptess? 4 wAoab s hT cpFReielr=ilenneaa 2ddtgriπ uo rTmalnu n/ig.ls mfgT 2le h io= set f h t5 fhle oeae nrfn moagdrctu mhcgle aul=l emlf a1or ea0rttt .oTri oem nsis a t:dakukeee l s tt ohT eg s rseaucvboitjneyd.cst toof cthoem fpolremteu olan.e full 16 20 ByM dyrastwici nrgo sseo mwiet hs i2m ppolein ctass es and cMouysnttiicn rgo tshee w liinthe s3, spoominet sresults can be found: c Hif og w= l1o0n?g is a pendulum which takes 3 seconds for one oscillation, Number of lines = 1 Number of lines = 3 C a Describe in words the elements of: i set A ii set B iii set C b Copy and complete the following statements: i A ∩ B = {...} ii A ∩ C = {...} iii B ∩ C = {...} iv A ∩ B ∩ C = {...} v A  B = {...} vi C  B = {...} 90 96 121 9781510421684_Ch10.indd 90 vi 30/12/17 5:55 PM9781510421684_Ch10.indd 96 30/12/17 5:55 PM9781510421684_Ch11.indd 121 30/12/17 6:44 PM 9781510421684.indb 6 22/02/18 1:54 PM Assessment For Cambridge IGCSE® Mathematics you will take two papers. If you are studying the Core syllabus, you will take Paper 1 and Paper 3. If you are studying the Extended syllabus you will take Paper 2 and Paper 4. You may use a scientific calculator for both papers. Length Type of questions Paper 1 (Core) 1 hour Short-answer questions Paper 2 (Extended) 1 hour 30 minutes Short-answer questions Paper 3 (Core) 2 hours Structured questions Paper 4 (Extended) 2 hours 30 minutes Structured questions Examination techniques Make sure you check the instructions on the question paper, the length of the paper and the number of questions you have to answer. In the case of Cambridge IGCSE Mathematics examinations you will have to answer every question as there will be no choice. Allocate your time sensibly between each question. Every year, good students let themselves down by spending too long on some questions and too little time (or no time at all) on others. Make sure you show your working to show how you’ve reached your answer. Command words The command words that may appear in your question papers are listed below. The command word used will relate to the context of the question. Command word What it means Calculate Work out from given facts, figures or information, generally using a calculator Construct* Make an accurate drawing Describe State the points of a topic/give characteristics and main features Determine Establish with certainty Explain Set out purposes or reasons/ make the relationships between things evident/ provide why and/or how and support with relevant evidence Give Produce an answer from a given source or recall/memory Plot Mark point(s) on a graph Show (that) Provide structured evidence that leads to a given result Sketch Make a simple freehand drawing showing the key features Work out Calculate from given facts, figures or information with or without the use of a calculator Write Give an answer in a specific form Write down Give an answer without significant working *Note: ‘construct’ is also used in the context of equations or expressions. When you construct an equation, you build it using information that you have been given or you have worked out. For example, you might construct an equation in the process of solving a word problem. vii 9781510421684.indb 7 22/02/18 1:54 PM From the authors Mathematics comes from the Greek word meaning knowledge or learning. Galileo Galilei (1564–1642) wrote “the universe cannot be read until we learn the language in which it is written. It is written in mathematical language”. Mathematics is used in science, engineering, medicine, art, finance etc., but mathematicians have always studied the subject for pleasure. They look for patterns in nature, for fun, as a game or a puzzle. A mathematician may find that his or her puzzle solving helps to solve ‘real life’ problems. But trigonometry was developed without a ‘real life’ application in mind, before it was then applied to navigation and many other things. The algebra of curves was not ‘invented’ to send a rocket to Jupiter. The study of mathematics is across all lands and cultures. A mathematician in Africa may be working with another in Japan to extend work done by a Brazilian in the USA. People in all cultures have tried to understand the world around them, and mathematics has been a common way of furthering that understanding, even in cultures which have left no written records. Each Topic in this text book has an introduction which tries to show how, over thousands of years, mathematical ideas have been passed from one culture to another. So when you are studying from this text book, remember that you are following in the footsteps of earlier mathematicians who were excited by the discoveries they had made. These discoveries changed our world. You may find some of the questions in this book difficult. It is easy when this happens to ask the teacher for help. Remember though that mathematics is intended to stretch the mind. If you are trying to get physically fit you do not stop as soon as things get hard. It is the same with mental fitness. Think logically. Try harder. In the end you are responsible for your own learning. Teachers and textbooks can only guide you. Be confident that you can solve that difficult problem. Ric Pimentel Terry Wall viii 9781510421684.indb 8 22/02/18 1:54 PM

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