Cambridge IGCSE Mathematics Core and Extended

560 Pages·2018·41.584 MB·English

by  Terry Wall, Ric Pimentel

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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
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Acknowledgements
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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.
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 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
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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
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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
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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.
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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
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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.
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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
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