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Contents
Abouttheleadauthorandconsultants ix
Introductionandoverview x
Acknowledgements xv
Preliminary topics
1
Functionsandrelations 1
1A Setnotationandsetsofnumbers . . . . . . . . . . . . . . . . 2
1B Identifyinganddescribingrelationsandfunctions . . . . . . . . 6
1C Implieddomainsandtypesoffunctions . . . . . . . . . . . . . 16
1D Combiningfunctions . . . . . . . . . . . . . . . . . . . . . . . 21
1E Powerfunctions . . . . . . . . . . . . . . . . . . . . . . . . . 26
1F Applicationsoffunctions . . . . . . . . . . . . . . . . . . . . . 31
ReviewofChapter1 . . . . . . . . . . . . . . . . . . . . . . . 35
2
Coordinategeometryandtransformations 41
2A Linearcoordinategeometry . . . . . . . . . . . . . . . . . . . 42
2B Translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2C Dilationsandreflections . . . . . . . . . . . . . . . . . . . . . 50
2D Combinationsoftransformations . . . . . . . . . . . . . . . . 54
2E Usingtransformationstosketchgraphs . . . . . . . . . . . . . 59
2F Transformationsofpowerfunctions . . . . . . . . . . . . . . . 62
2G Determiningtheruleforafunctionfromitsgraph . . . . . . . 66
ReviewofChapter2 . . . . . . . . . . . . . . . . . . . . . . . 68
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iv Contents
3
Polynomialfunctions 75
3A Quadraticfunctions. . . . . . . . . . . . . . . . . . . . . . . . 76
3B Determiningtheruleforaparabola . . . . . . . . . . . . . . . 85
3C Thelanguageofpolynomials . . . . . . . . . . . . . . . . . . . 90
3D Divisionandfactorisationofpolynomials . . . . . . . . . . . . 95
3E Thegeneralcubicfunction . . . . . . . . . . . . . . . . . . . . 103
3F Polynomialsofhigherdegree . . . . . . . . . . . . . . . . . . 107
3G Determiningtheruleforthegraphofapolynomial . . . . . . . 110
3H Solutionofliteralequationsandsystemsofequations . . . . . 116
ReviewofChapter3 . . . . . . . . . . . . . . . . . . . . . . . 120
4
Trigonometricfunctions 128
4A Measuringanglesindegreesandradians . . . . . . . . . . . . 129
4B Definingsine,cosineandtangent . . . . . . . . . . . . . . . . 131
4C FurthersymmetrypropertiesandthePythagoreanidentity . . . 138
4D Graphsofsineandcosine . . . . . . . . . . . . . . . . . . . . 140
4E Solutionoftrigonometricequations . . . . . . . . . . . . . . . 146
4F Sketchgraphsofy=asinn(t±ε)andy=acosn(t±ε) . . . . . 151
4G Sketchgraphsofy=asinn(t±ε)±bandy=acosn(t±ε)±b . 153
4H Determiningrulesforgraphsoftrigonometricfunctions . . . . 156
4I Thetangentfunction . . . . . . . . . . . . . . . . . . . . . . . 159
4J Applicationsoftrigonometricfunctions . . . . . . . . . . . . . 165
ReviewofChapter4 . . . . . . . . . . . . . . . . . . . . . . . 167
5
Revisionofpreliminarytopics 175
5A Technology-freequestions . . . . . . . . . . . . . . . . . . . . 175
5B Multiple-choicequestions . . . . . . . . . . . . . . . . . . . . 176
5C Extended-responsequestions . . . . . . . . . . . . . . . . . . 182
5D Degree-of-difficultyclassifiedquestions . . . . . . . . . . . . . 184
Unit 3
6
Exponentialandlogarithmicfunctions 189
6A Revisionofexponentialfunctions . . . . . . . . . . . . . . . . 190
6B Theexponentialfunctionf(x)=ex . . . . . . . . . . . . . . . . 196
6C Revisionofexponentialequations . . . . . . . . . . . . . . . . 200
6D Logarithmsandthelogarithmlaws . . . . . . . . . . . . . . . 202
6E Graphinglogarithmicfunctions . . . . . . . . . . . . . . . . . 209
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Contents v
6F Determiningrulesforgraphsofexponentialand
logarithmicfunctions . . . . . . . . . . . . . . . . . . . . . . . 213
6G Solvingequationsinvolvingexponentialand
logarithmicfunctions . . . . . . . . . . . . . . . . . . . . . . . 216
6H Applicationsofexponentialfunctions . . . . . . . . . . . . . . 222
6I Applicationsoflogarithmicfunctions . . . . . . . . . . . . . . 227
6J Modellingdatausingagraphicscalculator . . . . . . . . . . . 236
ReviewofChapter6 . . . . . . . . . . . . . . . . . . . . . . . 247
7
Refresherondifferentiation 253
7A Thederivative . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
7B Rulesfordifferentiation. . . . . . . . . . . . . . . . . . . . . . 259
7C Differentiatingxn wherenisanegativeinteger . . . . . . . . . 270
7D Thegraphofthederivativefunction . . . . . . . . . . . . . . . 273
ReviewofChapter7 . . . . . . . . . . . . . . . . . . . . . . . 278
8
Furtherdifferentiationandapplications 281
8A Thechainrule . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
8B Differentiatingrationalpowers . . . . . . . . . . . . . . . . . . 285
8C Differentiationofex . . . . . . . . . . . . . . . . . . . . . . . 288
8D Differentiationofthenaturallogarithmfunction . . . . . . . . 292
8E Differentiationoftrigonometricfunctions . . . . . . . . . . . . 294
8F Theproductrule . . . . . . . . . . . . . . . . . . . . . . . . . 298
8G Thequotientrule . . . . . . . . . . . . . . . . . . . . . . . . . 302
8H Tangentsandnormals . . . . . . . . . . . . . . . . . . . . . . 305
8I Ratesofchange. . . . . . . . . . . . . . . . . . . . . . . . . . 310
8J Motioninastraightline . . . . . . . . . . . . . . . . . . . . . 316
8K Stationarypoints . . . . . . . . . . . . . . . . . . . . . . . . . 322
8L Typesofstationarypoints . . . . . . . . . . . . . . . . . . . . 327
ReviewofChapter8 . . . . . . . . . . . . . . . . . . . . . . . 336
9
Anti-differentiation 346
9A Anti-differentiationofpolynomialfunctions . . . . . . . . . . . 347
9B Anti-differentiationofpowerfunctions . . . . . . . . . . . . . 352
9C Theanti-derivativeof(ax+b)r . . . . . . . . . . . . . . . . . . 354
9D Theanti-derivativeofekx . . . . . . . . . . . . . . . . . . . . . 357
9E Anti-differentiationoftrigonometricfunctions . . . . . . . . . 358
9F Furtheranti-differentiationtechniques . . . . . . . . . . . . . . 360
9G Applicationsofanti-differentiationtomotionina
straightline . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363
ReviewofChapter9 . . . . . . . . . . . . . . . . . . . . . . . 367
C a m b r id g e S e n i o r M a t h s f o r Q u e e n s l a n d ISBN 978-1-108-45164-2 © Evans et al. 2019 Cambridge University Press
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vi Contents
10
Integration 372
10A Estimatingtheareaunderagraph . . . . . . . . . . . . . . . . 373
10B Findingtheexactarea: thedefiniteintegral . . . . . . . . . . . 378
10C Signedarea . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383
10D Integrationofmorefamiliesoffunctions . . . . . . . . . . . . 390
10E Furtherintegrationtechniques . . . . . . . . . . . . . . . . . . 394
10F Theareaofaregionbetweentwocurves . . . . . . . . . . . . 397
10G Applicationsofintegration . . . . . . . . . . . . . . . . . . . . 402
10H Theareaunderagraphasthelimitofasum . . . . . . . . . . 409
ReviewofChapter10 . . . . . . . . . . . . . . . . . . . . . . 411
11
RevisionofUnit3 421
11A Technology-freequestions . . . . . . . . . . . . . . . . . . . . 421
11B Multiple-choicequestions . . . . . . . . . . . . . . . . . . . . 425
11C Extended-responsequestions . . . . . . . . . . . . . . . . . . 432
11D Degree-of-difficultyclassifiedquestions . . . . . . . . . . . . . 436
OnlineassessmentpracticeintheInteractiveTextbookandOnlineTeachingSuite
IA1: Practiceproblem-solvingandmodellingtask
IA2: PracticeinternalexaminationonUnit3
Unit 4
12
Thesecondderivativeandapplications 440
12A Thesecondderivativeandacceleration . . . . . . . . . . . . . 441
12B Usingthesecondderivativeingraphsketching . . . . . . . . . 445
12C Absolutemaximumandminimumvalues . . . . . . . . . . . . 455
12D Optimisationproblems . . . . . . . . . . . . . . . . . . . . . . 458
ReviewofChapter12 . . . . . . . . . . . . . . . . . . . . . . 469
13
Trigonometryusingthesineandcosinerules 479
13A Reviewingtrigonometry . . . . . . . . . . . . . . . . . . . . . 480
13B Thesinerule . . . . . . . . . . . . . . . . . . . . . . . . . . . 486
13C Thecosinerule . . . . . . . . . . . . . . . . . . . . . . . . . . 490
13D Theareaofatriangle . . . . . . . . . . . . . . . . . . . . . . . 494
13E Anglesofelevation,anglesofdepressionandbearings . . . . . 497
13F Problemsinthreedimensions . . . . . . . . . . . . . . . . . . 501
13G Anglesbetweenplanesandmorecomplex3Dproblems . . . . 505
ReviewofChapter13 . . . . . . . . . . . . . . . . . . . . . . 509
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Contents vii
14
Refresheronprobabilityanddiscreterandomvariables 515
14A Samplespacesandprobability . . . . . . . . . . . . . . . . . . 516
14B Conditionalprobabilityandindependence . . . . . . . . . . . . 526
14C Discreterandomvariables . . . . . . . . . . . . . . . . . . . . 534
14D Expectedvalue,varianceandstandarddeviation . . . . . . . . 541
ReviewofChapter14 . . . . . . . . . . . . . . . . . . . . . . 552
15
Bernoullisequencesandthebinomialdistribution 559
15A IntroductiontoBernoullisequencesandthe
binomialdistribution . . . . . . . . . . . . . . . . . . . . . . . 560
15B Thegraph,expectationandvarianceofa
binomialdistribution . . . . . . . . . . . . . . . . . . . . . . . 567
15C Findingthesamplesize . . . . . . . . . . . . . . . . . . . . . . 571
15D Proofsfortheexpectationandvariance . . . . . . . . . . . . . 574
ReviewofChapter15 . . . . . . . . . . . . . . . . . . . . . . 576
16
Continuousrandomvariables 581
16A Introductiontocontinuousrandomvariables . . . . . . . . . . 582
16B Meanandmedianforacontinuousrandomvariable . . . . . . 594
16C Measuresofspread . . . . . . . . . . . . . . . . . . . . . . . . 602
16D Propertiesofmeanandvariance . . . . . . . . . . . . . . . . . 607
16E Cumulativedistributionfunctions . . . . . . . . . . . . . . . . 610
ReviewofChapter16 . . . . . . . . . . . . . . . . . . . . . . 614
17
Thenormaldistribution 622
17A Thenormaldistribution . . . . . . . . . . . . . . . . . . . . . . 623
17B Standardisation . . . . . . . . . . . . . . . . . . . . . . . . . . 629
17C Determiningnormalprobabilities. . . . . . . . . . . . . . . . . 634
17D Solvingproblemsusingthenormaldistribution . . . . . . . . . 640
17E Thenormalapproximationtothebinomialdistribution . . . . . 645
ReviewofChapter17 . . . . . . . . . . . . . . . . . . . . . . 648
18
Samplingandestimation 654
18A Populationsandsamples . . . . . . . . . . . . . . . . . . . . . 655
18B Theexactdistributionofthesampleproportion . . . . . . . . . 664
18C Approximatingthedistributionofthesampleproportion . . . . 672
18D Confidenceintervalsforthepopulationproportion . . . . . . . 678
ReviewofChapter18 . . . . . . . . . . . . . . . . . . . . . . 688
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viii Contents
19
RevisionofUnit4 694
19A Technology-freequestions . . . . . . . . . . . . . . . . . . . . 694
19B Multiple-choicequestions . . . . . . . . . . . . . . . . . . . . 698
19C Extended-responsequestions . . . . . . . . . . . . . . . . . . 704
19D Degree-of-difficultyclassifiedquestions . . . . . . . . . . . . . 709
20
RevisionofUnits3&4 716
20A Technology-freequestions . . . . . . . . . . . . . . . . . . . . 716
20B Multiple-choicequestions . . . . . . . . . . . . . . . . . . . . 720
20C Extended-responsequestions . . . . . . . . . . . . . . . . . . 724
20D Degree-of-difficultyclassifiedquestions . . . . . . . . . . . . . 738
A
AppendixA:Countingmethodsandthebinomialtheorem 743
A1 Countingmethods . . . . . . . . . . . . . . . . . . . . . . . . 743
A2 Summationnotation . . . . . . . . . . . . . . . . . . . . . . . 746
A3 Thebinomialtheorem . . . . . . . . . . . . . . . . . . . . . . 747
OnlineassessmentpracticeintheInteractiveTextbookandOnlineTeachingSuite
IA3: PracticeinternalexaminationonUnit4
EA: PracticeexternalexaminationonUnits3and4
Glossary 750
Answers 761
OnlineAppendixB:Guidestousingtechnology
TheseonlineguidesareaccessedthroughtheInteractiveTextbookorPDFTextbook
B1 OnlineguidetotheTI-NspireCXNon-CASgraphicscalculator
B2 OnlineguidetotheTI-84PlusCEgraphicscalculator
B3 OnlineguidetotheCasiofxCG20AUandCasiofxCG50AUgraphics
calculators
B4 OnlineguidetotheDesmosgraphingcalculator
Note: AprintablecopyoftheQCAAFormulasheetisavailableintheInteractive
Textbook
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About the
lead author and
consultants
About the lead author
MichaelEvanswasaconsultanttoACARAonthewritingoftheAustralianCurriculum
onwhichthenewQueenslandsyllabusisbased. HeisaconsultantwiththeAustralian
MathematicalSciencesInstitute,andiscoordinatingauthoroftheICE-EM7–10seriesalso
publishedbyCambridge.
HehasalsobeenactiveintheAustralianMathematicsTrust,beinginvolvedwiththewriting
ofenrichmentmaterialandcompetitionquestions.
Hehasmanyyears’experienceasaChiefExaminerandChairpersonofexaminationpanels.
About the consultants
ChicriMaksoudisSeniorMathematicsTeacheratBrisbaneBoys’College
SteveSissonisCurriculumLeader–MathematicsatRedeemerLutheranCollege,
Rochedale,QLD
TrevorRedmondisHeadofMathematicsatSomervilleHouse,SouthBrisbane
RayMinnsisHeadofMathematicsatNorthpineChristianCollege,Dakabin,QLD
C a m b r id g e S e n i o r M a t h s f o r Q u e e n s l a n d ISBN 978-1-108-45164-2 © Evans et al. 2019 Cambridge University Press
M a t h e m a t i c a l M e t h o d s 3 & 4 Photocopying is restricted under law and this material must not be transferred to another party.
Introduction and
overview
CambridgeSeniorMathematicsforQueenslandMathematicalMethodsUnits3&4provides
completeandclosecoverageoftheQCAAsyllabustobeimplementedinYear12from
2020. Itsfourcomponents—theprintbook,downloadablePDFtextbook,onlineInteractive
Textbook(ITB)andOnlineTeachingResource(OTS),bothpoweredbytheHOTmaths
platform—containahugerangeofresources,includingworkedsolutionsavailableto
schoolsinasinglepackageatoneconvenientprice(theOTSisincludedwithclassadoptions,
conditionsapply). Therearenoextrasubscriptionsorper-studentchargestopay.
Preliminarytopics(reviewofUnits1&2): Thefirstfourchapterscanbeconsideredasa
reviewofUnits1&2: Chapter1Functionsandrelations,Chapter2Coordinategeometry
andtransformations,Chapter3PolynomialfunctionsandChapter4Trigonometricfunctions.
ThetopicscoveredinthesechaptersareimportantforUnits3&4andofcoursemaybe
examinedatyear12. Youmaychoosetocompletethesechapterspriortothebeginningof
Year12.
Inaddition,two‘refresher’chaptersareprovided: Chapter7Refresherondifferentiationand
Chapter14Refresheronprobabilityanddiscreterandomvariables. Itisrecommendedthat
thesebedonejustbeforethechaptersforwhichtheyarepreparation.
Tohelpdecidewhetheranystudentscanbeexemptedfromdoingthepreliminarytopicsand
refresherchapters,themultiple-choicequestionsectionsfromtheirchapterreviewsaresetup
intheOnlineTeachingSuitetoprovidetheoptionofbeingusedasdiagnostictestsforthis
purpose.
Degreeofdifficultyclassificationofquestions: intheexercises,questionsareclassified
assimplefamiliar SF,complexfamiliar CF,andcomplexunfamiliar CU questions. The
revisionchaptersdescribedbelowalsocontainmodelquestionsforeachofthesecategories,
andtestsarealsoprovidedintheteacherresources,madeupofsuchcategorisedmodel
questions.
ThreerevisionchaptersofmaterialcoveredinUnits3and4: Thesechapterscontain
sectionsonTechnology-freequestions,Multiplechoicequestions,Extended-response
questions,andDegreeofdifficultyclassificationofquestions. Thefirstrevisionchapter
occursattheendofUnit3,thesecondattheendofUnit4andthereisafinalrevisionchapter
thatwillhelpwithrevisionfortheexternalexamination
Calculatorguidance: ThroughoutthebookthereisguidancefortheuseoftheTI-Nspire
CXnon-CASandtheCasiofxCG20AUandfxCG50AUgraphicscalculatorsforthesolution
ofproblems. GuidanceontheTI-84PlusCEisincludedintheinteractivetextbook,accessed
viaiconsnexttotheTI-Nspireboxes. Therearealsoonlineguidesforthegeneraluseofeach
ofthesecalculators.
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M a t h e m a t i c a l M e t h o d s 3 & 4 Photocopying is restricted under law and this material must not be transferred to another party.
