IGCSE Cambridge International Mathematics

752 Pages·2008·14.73 MB·English

Haese and Harris Publications
specialists in mathematics publishing
Endorsed by University of Cambridge International Examinations
IGCSE
Cambridge
International
Mathematics
(0607)
Extended
Keith Black
Alison Ryan
Michael Haese
Robert Haese
Sandra Haese
Mark Humphries
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IGCSECAMBRIDGEINTERNATIONALMATHEMATICS(0607)
KeithBlack B.Sc.(Hons.),Dip.Ed.
AlisonRyan B.Sc.,M.Ed.
MichaelHaese B.Sc.(Hons.),Ph.D.
RobertHaese B.Sc.
SandraHaese B.Sc.
MarkHumphries B.Sc.(Hons.)
Haese&HarrisPublications
3FrankCollopyCourt,AdelaideAirport, SA5950,AUSTRALIA
Telephone: [phone], Fax: [phone]
Email: [email protected]
Web: www.haeseandharris.com.au
NationalLibraryofAustraliaCardNumber&ISBN 978-1-921500-04-6
©Haese&HarrisPublications2009
PublishedbyRaksarNomineesPtyLtd
3FrankCollopyCourt,AdelaideAirport, SA5950,AUSTRALIA
FirstEdition 2009
CartoonartworkbyJohnMartin.ArtworkandcoverdesignbyPiotrPoturaj.
FractalartworkonthecovercopyrightbyJarosławWierny,www.fractal.art.pl
ComputersoftwarebyDavidPurton,TroyCruickshankandThomasJansson.
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This textbook and its accompanying CD have been endorsed by University of Cambridge International
Examinations(CIE).TheyhavebeendevelopedindependentlyoftheInternationalBaccalaureateOrganization
(IBO)andarenotconnectedwithorendorsedby,theIBO.
This book is copyright. Except as permitted by the Copyright Act (any fair dealing for the purposes of
private study, research, criticism or review), no part of this publication may be reproduced, stored in a
retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying,
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Publications.
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oftheCopyrightAct,thelawrequiresthattheeducationalinstitutionorthebodythatadministersithasgiven
a remuneration notice to CopyrightAgency Limited (CAL). For information, contact the CopyrightAgency
Limited.
Acknowledgements:ThepublishersacknowledgethecooperationofOxfordUniversityPress,Australia,forthe
reproduction of material originally published in textbooks produced in association with
Haese&HarrisPublications.
Whileeveryattempthasbeenmadetotraceandacknowledgecopyright,theauthorsandpublishersapologisefor
any accidental infringement where copyright has proved untraceable. They would be pleased to come to a
suitableagreementwiththerightfulowner.
Disclaimer:All the internet addresses (URL’s) given in this book were valid at the time of printing. While
the authors and publisher regret any inconvenience that changes of address may cause readers, no
responsibilityforanysuchchangescanbeacceptedbyeithertheauthorsorthepublisher.
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FOREWORD
This book has been written to cover the ‘IGCSE Cambridge International Mathematics (0607) Extended’
courseoveratwo-yearperiod.
ThenewcoursewasdevelopedbyUniversityofCambridgeInternationalExaminations(CIE)inconsultation
with teachers in international schools around the world. It has been designed for schools that want their
mathematics teaching to focus more on investigations and modelling, and to utilise the powerful technology
ofgraphicscalculators.
Thecoursespringsfromtheprinciplesthatstudentsshoulddevelopagoodfoundationofmathematicalskills
and that they should learn to develop strategies for solving open-ended problems. It aims to promote a
positive attitude towards Mathematics and a confidence that leads to further enquiry. Some of the schools
consulted by CIE were IB schools and as a result, Cambridge International Mathematics integrates
exceptionallywellwiththeapproachtotheteachingofMathematicsinIBschools.
This book is an attempt to cover, in one volume, the content outlined in the Cambridge International
Mathematics (0607) syllabus. References to the syllabus are made throughout but the book can be used as a
full course in its own right, as a preparation for GCE Advanced Level Mathematics or IB Diploma
Mathematics, for example.The book has been endorsed by CIE but it has been developed independently of
theIndependentBaccalaureateOrganizationandisnotconnectedwith,orendorsedby,theIBO.
To reflect the principles on which the new course is based, we have attempted to produce a book and CD
package that embraces technology, problem solving, investigating and modelling, in order to give students
different learning experiences.There are non-calculator sections as well as traditional areas of mathematics,
especially algebra.An introductory section ‘Graphics calculator instructions’appears on p. 11. It is intended
as a basic reference to help students who may be unfamiliar with graphics calculators. Two chapters of
‘assumedknowledge’areaccessiblefromtheCD:‘Number’and‘Geometryandgraphs’(seepp.29and30).
Theycanbeprintedforthosewhowanttoensurethattheyhavetheprerequisitelevelsofunderstandingfor
thecourse.Toreflectoneofthemainaimsofthenewcourse,thelasttwochaptersinthebookaredevotedto
multi-topic questions, and investigations and modelling. Review exercises appear at the end of each chapter
with some ‘Challenge’ questions for the more able student. Answers are given at the end of the book,
followedbyanindex.
The interactive CD contains Self Tutor software (see p. 5), geometry and graphics software,
demonstrationsandsimulations,andthetwoprintablechaptersonassumedknowledge.TheCDalsocontains
thetextofthebooksothatstudentscanloaditonahomecomputerandkeepthetextbookatschool.
TheCambridgeInternationalMathematicsexaminationsareintheformofthreepapers:oneanon-calculator
paper, another requiring the use of a graphics calculator, and a third paper containing an investigation and a
modellingquestion.Alloftheseaspectsofexaminingareaddressedinthebook.
Thebookcanbeusedasaschemeofworkbutitisexpectedthattheteacherwillchoosetheorderoftopics.
Thereareafewoccasionswhereaquestioninanexercisemayrequiresomethingdonelaterinthebookbut
this has been kept to a minimum. Exercises in the book range from routine practice and consolidation of
basicskills,toproblemsolvingexercisesthatarequitedemanding.
In this changing world of mathematics education, we believe that the contextual approach shown in this
book, with the associated use of technology, will enhance the students’ understanding, knowledge and
appreciationofmathematics,anditsuniversalapplication.
Wewelcomeyourfeedback.
Email: [email protected]
Web: www.haeseandharris.com.au KB,AR,PMH,RCH,SHH,MH
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ACKNOWLEDGEMENTS
The authors and publishers would like to thank University of Cambridge International Examinations (CIE)
for their assistance and support in the preparation of this book. Exam questions from past CIE exam papers
arereproducedbypermissionoftheUniversityofCambridgeLocalExaminationsSyndicate.TheUniversity
of Cambridge Local Examinations Syndicate bears no responsibility for the example answers to questions
takenfromitspastquestionpaperswhicharecontainedinthispublication.
Inadditionwewouldliketothanktheteacherswhoofferedtoreadproofsandwhogaveadviceandsupport:
SimonBullock,PhilipKurbis,RichardHenry,JohnnyRamesar,AlanDaykin,NigelWheeler,YenerBalkaya,
andspecialthanksisduetoFranO'Connorwhogotusstarted.
The publishers wish to make it clear that acknowledging these teachers, does not imply any endorsement of
thisbookbyanyofthem,andallresponsibilityforthecontentrestswiththeauthorsandpublishers.
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USING THE INTERACTIVE CD
TheinteractiveStudentCDthatcomeswiththisbookisdesignedforthosewho
wanttoutilisetechnologyinteachingandlearningMathematics.
TheCDiconthatappearsthroughoutthebookdenotesanactivelinkontheCD.
SimplyclickontheiconwhenrunningtheCDtoaccessalargerangeofinteractive
featuresthatincludes:
• spreadsheets
(cid:129) printableworksheets
INTERACTIVELINK
(cid:129) graphingpackages
(cid:129) geometrysoftware
(cid:129) demonstrations
(cid:129) simulations
(cid:129) printablechapters
(cid:129) SELFTUTOR
Forthosewhowanttoensuretheyhavetheprerequisitelevelsofunderstandingforthisnewcourse,printable
chaptersofassumedknowledgeareprovidedforNumber(seep.29) andGeometryandGraphs(seep.30).
SELF TUTOR is an exciting feature of this book.
The Self Tutor icon on each worked example denotes an active link on the CD.
Simply ‘click’ on the Self Tutor (or anywhere in the example box) to access the worked
example, with a teacher’s voice explaining each step necessary to reach the answer.
Play any line as often as you like. See how the basic processes come alive using movement and
colour on the screen.
Ideal for students who have missed lessons or need extra help.
Example 8 Self Tutor
Adiehasthenumbers0,0,1,1,4and5.Itisrolledtwice.Illustratethesamplespaceusing
a2-Dgrid.Hencefindtheprobabilityofgetting:
a a total of 5 b two numberswhich are the same.
2-Dgrid roll1 5 There are 6£6=36 possible outcomes.
4 a P(total of 5) = 8 fthose with a g
1 36
1
0
b P(same numbers) = 10 fthose circled g
0 36
roll2
0 0 1 1 4 5
SeeChapter25,Probability,p.516
GRAPHICSCALCULATORS
The course assumes that each student will have a graphics calculator. An introductory section ‘Graphics
calculatorinstructions’appearsonp.11.Tohelpgetstudentsstarted,thesectionincludessomebasicinstructions
fortheTexasInstrumentsTI-84PlusandtheCasiofx-9860Gcalculators.
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SYMBOLS AND NOTATION USED IN THIS BOOK
N thesetofpositiveintegersandzero, > isgreaterthan
f0,1,2,3,......g ¸or> isgreaterthanorequalto
Z thesetofintegers, f0,§1,§2,§3,......g < islessthan
Z+ thesetofpositiveintegers, f1,2,3,......g ·or6 islessthanorequalto
Q thesetofrationalnumbers un thenthtermofasequenceorseries
Q+ thesetofpositiverationalnumbers, f : x7!y f isafunctionunderwhichxismappedtoy
fxjx>0, x2Qg
f(x) theimageofxunderthefunctionf
R thesetofrealnumbers f¡1 theinversefunctionofthefunctionf
R+ thesetofpositiverealnumbers,
fxjx>0, x2Rg logax logarithmtothebaseaofx
sin,cos,tan thecircularfunctions
fx1,x2,....g thesetwithelements x1,x2,.....
A(x,y) thepointAintheplanewithCartesian
n(A) thenumberofelementsinthefinitesetA
coordinatesxandy
(
fxj...... thesetofallxsuchthat thelinesegmentwithendpointsAandB
2 isanelementof AB thedistancefromAtoB
thelinecontainingpointsAandB
2= isnotanelementof
b
? or f g theempty(null)set A theangleatA
b
U theuniversalset CAB theanglebetweenCAandAB
[ union ¢ABC thetrianglewhoseverticesareA,BandC
\ intersection v thevectorv
¡!
µ isasubsetof AB thevectorrepresentedinmagnitudeanddirection
bythedirectedlinesegmentfromAtoB
½ isapropersubsetof
jaj themagnitudeofvectora
A0 thecomplementofthesetA
¡! ¡!
an1, pna atothepowerofpn1, nthrootofa Pj(AAB)j pthreobmabagilnitiytuodfeeovfeAntBA
(if a>0 then na>0)
P(A0) probabilityoftheevent“notA”
a12,pa atothepower 12,psquarerootofa x1,x2,.... observationsofavariable
(if a>0 then a>0)
f1,f2,.... frequencieswithwhichtheobservations
jxj tnhemodulusorabsolutevalueofx,thatis x1,x2,x3,..... occur
xforx>0, x2R x meanofthevalues x1,x2,....
¡xforx<0, x2R
§f sumofthefrequencies f1,f2,....
´ identity or isequivalentto
r Pearson’scorrelationcoefficient
¼ isapproximatelyequalto r2 coefficientofdetermination
»= iscongruentto
k isparallelto
isperpendicularto
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Table of contents 7
TABLE OF CONTENTS
J Expressionswithfourterms 48
K Factorising xX(cid:2)+(cid:2)bx(cid:2)+(cid:2)c 49
L Splittingthemiddleterm 51
M Miscellaneousfactorisation 54
SYMBOLSANDNOTATION Reviewset1A 55
Reviewset1B 56
USEDINTHISBOOK 6
2 SETS 57
GRAPHICSCALCULATOR
INSTRUCTIONS 11 A Setnotation 57
B Specialnumbersets 60
A Basiccalculations 12
C Intervalnotation 61
B Basicfunctions 13
D Venndiagrams 63
C Secondaryfunctionandalphakeys 17
E Unionandintersection 65
D Memory 17
F Problemsolving 69
E Lists 19
Reviewset2A 72
F Statisticalgraphs 21
Reviewset2B 73
G Workingwithfunctions 22
H Twovariableanalysis 26
3 ALGEBRA
(EQUATIONSANDINEQUALITIES) 75
ASSUMEDKNOWLEDGE(NUMBER) 29
A Solvinglinearequations 75
A Numbertypes CD
B Solvingequationswithfractions 80
B Operationsandbrackets CD
C Formingequations 83
C HCFandLCM CD
D Problemsolvingusingequations 85
D Fractions CD
E Powerequations 87
E Powersandroots CD
F Interpretinglinearinequalities 88
F Ratioandproportion CD
G Solvinglinearinequalities 89
G Numberequivalents CD
Reviewset3A 91
H Roundingnumbers CD
Reviewset3B 92
I Time CD
4 LINES,ANGLESANDPOLYGONS 93
ASSUMEDKNOWLEDGE
(GEOMETRYANDGRAPHS) 30 A Angleproperties 93
B Triangles 98
A Angles CD
C Isoscelestriangles 100
B Linesandlinesegments CD
D Theinterioranglesofapolygon 103
C Polygons CD
E Theexterioranglesofapolygon 106
D Symmetry CD
Reviewset4A 107
E Constructingtriangles CD
Reviewset4B 109
F Congruence CD
G Interpretinggraphsandtables CD
5 GRAPHS,CHARTSANDTABLES 111
1 ALGEBRA A Statisticalgraphs 112
(EXPANSIONANDFACTORISATION) 31 B Graphswhichcomparedata 116
C Usingtechnologytographdata 119
A Thedistributivelaw 32
Reviewset5A 120
B Theproduct(a+b)(c+d) 33
Reviewset5B 122
C Differenceoftwosquares 35
D Perfectsquaresexpansion 37
6 EXPONENTSANDSURDS 123
E Furtherexpansion 39
F Algebraiccommonfactors 40 A Exponentorindexnotation 123
G Factorisingwithcommonfactors 42 B Exponentorindexlaws 126
H Differenceoftwosquaresfactorisation 45 C Zeroandnegativeindices 129
I Perfectsquaresfactorisation 47 D Standardform 131
E Surds 134
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8 Table of contents
F Propertiesofsurds 137 C Capacity 245
G Multiplicationofsurds 139 D Mass 248
H Divisionbysurds 142 E Compoundsolids 249
Reviewset6A 143 Reviewset11A 253
Reviewset6B 145 Reviewset11B 254
7 FORMULAEANDSIMULTANEOUS 12 COORDINATEGEOMETRY 255
EQUATIONS 147
A Plottingpoints 256
A Formulasubstitution 148 B Distancebetweentwopoints 258
B Formularearrangement 150 C Midpointofalinesegment 261
C Formuladerivation 153 D Gradientofalinesegment 263
D Moredifficultrearrangements 155 E Gradientofparalleland
E Simultaneousequations 158 perpendicularlines 267
F Problemsolving 164 F Usingcoordinategeometry 270
Reviewset7A 166 Reviewset12A 272
Reviewset7B 167 Reviewset12B 273
8 THETHEOREMOFPYTHAGORAS 169 13 ANALYSISOFDISCRETEDATA 275
A Pythagoras’theorem 170 A Variablesusedinstatistics 277
B TheconverseofPythagoras’theorem 176 B Organisinganddescribingdiscretedata 278
C Problemsolving 177 C Thecentreofadiscretedataset 282
D Circleproblems 181 D Measuringthespreadofdiscretedata 285
E Three-dimensionalproblems 185 E Datainfrequencytables 288
Reviewset8A 187 F Groupeddiscretedata 290
Reviewset8B 188 G Statisticsfromtechnology 292
Reviewset13A 293
9 MENSURATION Reviewset13B 295
(LENGTHANDAREA) 191
14 STRAIGHTLINES 297
A Length 192
B Perimeter 194 A Verticalandhorizontallines 297
C Area 196 B Graphingfromatableofvalues 299
D Circlesandsectors 201 C Equationsoflines
Reviewset9A 206 (gradient-interceptform) 301
Reviewset9B 207 D Equationsoflines(generalform) 304
E Graphinglinesfromequations 307
10 TOPICSINARITHMETIC 209 F Linesofsymmetry 308
Reviewset14A 310
A Percentage 209
Reviewset14B 311
B Profitandloss 211
C Simpleinterest 214
15 TRIGONOMETRY 313
D Reversepercentageproblems 217
E Multipliersandchainpercentage 218 A Labellingsidesofarightangledtriangle 314
F Compoundgrowth 222 B Thetrigonometricratios 316
G Speed,distanceandtime 224 C Problemsolving 322
H Travelgraphs 226 D Thefirstquadrantoftheunitcircle 327
Reviewset10A 228 E Truebearings 330
Reviewset10B 229 F 3-dimensionalproblemsolving 331
Reviewset15A 336
11 MENSURATION Reviewset15B 337
(SOLIDSANDCONTAINERS) 231
16 ALGEBRAICFRACTIONS 339
A Surfacearea 231
B Volume 239 A Simplifyingalgebraicfractions 339
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Table of contents 9
B Multiplyinganddividingalgebraic E Graphsofquadraticfunctions 431
fractions 344 F Axesintercepts 438
C Addingandsubtractingalgebraic G Lineofsymmetryandvertex 441
fractions 346 H Findingaquadraticfunction 445
D Morecomplicatedfractions 348 I Usingtechnology 446
Reviewset16A 351 J Problemsolving 447
Reviewset16B 352 Reviewset21A 451
Reviewset21B 453
17 CONTINUOUSDATA 353
22 TWOVARIABLEANALYSIS 455
A Themeanofcontinuousdata 354
B Histograms 355 A Correlation 456
C Cumulativefrequency 359 B Lineofbestfitbyeye 459
Reviewset17A 364 C Linearregression 461
Reviewset17B 365 Reviewset22A 466
Reviewset22B 467
18 SIMILARITY 367
23 FURTHERFUNCTIONS 469
A Similarity 367
B Similartriangles 370 A Cubicfunctions 469
C Problemsolving 373 B Inversefunctions 473
D Areaandvolumeofsimilarshapes 376 C Usingtechnology 475
Reviewset18A 380 D Tangentstocurves 480
Reviewset18B 381 Reviewset23A 481
Reviewset23B 481
19 INTRODUCTIONTOFUNCTIONS 383
24 VECTORS 483
A Mappingdiagrams 383
B Functions 385 A Directedlinesegmentrepresentation 484
C Functionnotation 389 B Vectorequality 485
D Compositefunctions 391 C Vectoraddition 486
E Reciprocalfunctions 393 D Vectorsubtraction 489
F Theabsolutevaluefunction 395 E Vectorsincomponentform 491
Reviewset19A 398 F Scalarmultiplication 496
Reviewset19B 399 G Parallelvectors 497
H Vectorsingeometry 499
20 TRANSFORMATIONGEOMETRY 401 Reviewset24A 501
Reviewset24B 503
A Translations 402
B Rotations 404
25 PROBABILITY 505
C Reflections 406
D Enlargementsandreductions 408 A Introductiontoprobability 506
E Stretches 410 B Estimatingprobability 507
F Transformingfunctions 413 C Probabilitiesfromtwo-waytables 510
G Theinverseofatransformation 416 D Expectation 512
H Combinationsoftransformations 417 E Representingcombinedevents 513
Reviewset20A 419 F Theoreticalprobability 515
Reviewset20B 420 G Compoundevents 519
H Usingtreediagrams 522
21 QUADRATICEQUATIONSAND I Samplingwithandwithoutreplacement 524
FUNCTIONS 421 J Mutuallyexclusiveand
non-mutuallyexclusiveevents 527
A Quadraticequations 422
K Miscellaneousprobabilityquestions 528
B TheNullFactorlaw 423
Reviewset25A 530
C Thequadraticformula 427
Reviewset25B 531
D Quadraticfunctions 429
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10 Table of contents
26 SEQUENCES 533 C Rulesforlogarithms 629
D Logarithmsinbase10 630
A Numbersequences 534
E Exponentialandlogarithmicequations 634
B Algebraicrulesforsequences 535
Reviewset31A 636
C Geometricsequences 537
Reviewset31B 637
D Thedifferencemethodforsequences 539
Reviewset26A 544
32 INEQUALITIES 639
Reviewset26B 545
A Solvingonevariableinequalitieswith
27 CIRCLEGEOMETRY 547 technology 639
B Linearinequalityregions 641
A Circletheorems 547
C Integerpointsinregions 644
B Cyclicquadrilaterals 556
D Problemsolving(Extension) 645
Reviewset27A 561
Reviewset32A 647
Reviewset27B 562
Reviewset32B 648
28 EXPONENTIALFUNCTIONSAND
33 MULTI-TOPICQUESTIONS 649
EQUATIONS 565
A Rationalexponents 566 34 INVESTIGATIONANDMODELLING
B Exponentialfunctions 568 QUESTIONS 661
C Exponentialequations 570
A Investigationquestions 661
D Problemsolvingwith
B Modellingquestions 669
exponentialfunctions 573
E Exponentialmodelling 576
ANSWERS 673
Reviewset28A 577
Reviewset28B 578 INDEX 752
29 FURTHERTRIGONOMETRY 579
A Theunitcircle 579
B Areaofatriangleusingsine 583
C Thesinerule 585
D Thecosinerule 588
E Problemsolvingwiththesine
andcosinerules 591
F Trigonometrywithcompoundshapes 593
G Trigonometricgraphs 595
H Graphsofy(cid:2)=(cid:2)a(cid:2)sin(bx)andy(cid:2)=(cid:2)a(cid:2)cos(bx) 599
Reviewset29A 601
Reviewset29B 602
30 VARIATIONAND
POWERMODELLING 605
A Directvariation 606
B Inversevariation 612
C Variationmodelling 615
D Powermodelling 619
Reviewset30A 622
Reviewset30B 623
31 LOGARITHMS 625
A Logarithmsinbasea 625
B Thelogarithmicfunction 627
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