Specialist Mathematics

767 Pages·48.253 MB·English

by  Michael Evans, Douglas Wallace, Kay Lipson, David Treeby

e

s c
n a
l
a
l
a
v
y
E W
b
n
l o e
e s
e
s
a
a
p r
h gl i T
c L
u d
i
M o y i
v
D a
a
K
D
S
p
e
M
c
a i
t
a
l
h i
e s
t
m
a
t
i
c
2
s

&
Ca
1 m
b
ts Senior ridge
i M
n at
h
e
U
A m
ustr atic
C alia s INCLUDES INTERACTIVE
ur n TEXTBOOK POWERED BY
ri
c CAMBRIDGE HOTMATHS
ul
u
m
/
V
C
E
Cambridge Senior Maths AC/VC E ISBN 978-1-107-56765-8 © Evans et al. 2016 Cambridge University Press
Specialist Mathematics 1&2 Photocopying is restricted under law and this material must not be transferred to another party. Updated May 2021
e

s c
n a
l
a
l
a
v
y
E W
b
n
l o e
e s
e
s
a
a
p r
h gl i T
c L
u d
i
M o y i
v
D a
a
K
D
S
p
e
M
c
a i
t
a
l
h i
e s
t
m
a
t
i
c
2
s

&
Ca
1 m
b
ts Senior ridge
i M
n at
h
e
U
A m
ustr atic
C alia s INCLUDES INTERACTIVE
ur n TEXTBOOK POWERED BY
ri
c CAMBRIDGE HOTMATHS
ul
u
m
/
V
C
E
Cambridge Senior Maths AC/VC E ISBN 978-1-107-56765-8 © Evans et al. 2016 Cambridge University Press
Specialist Mathematics 1&2 Photocopying is restricted under law and this material must not be transferred to another party. Updated May 2021
UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom
OneLibertyPlaza,20thFloor,NewYork,NY10006,USA
477WilliamstownRoad,PortMelbourne,VIC3207,Australia
314–321,3rdFloor,Plot3,SplendorForum,JasolaDistrictCentre,NewDelhi–110025,India
103PenangRoad,#05–06/07,VisioncrestCommercial,Singapore238467
CambridgeUniversityPressispartoftheUniversityofCambridge.
ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof
education,learningandresearchatthehighestinternationallevelsofexcellence.
www.cambridge.org
©MichaelEvans,DouglasWallace,KayLipson,DavidTreeby2016
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstprinted2016
20 19 18 17 16 15 14 13 12 11 10 9 8 7
CoverdesignedbyLoupeDesign
TypesetbyJanePitkethlyanddiacriTech
PrintedinChinabyC&COffsetPrintingCo.Ltd
Acataloguerecordforthisbookisavailablefrom
theNationalLibraryofAustraliaatwww.nla.gov.au
ISBN978-1-107-56765-8Paperback
Additionalresourcesforthispublicationatwww.cambridge.edu.au/GO
Reproductionandcommunicationforeducationalpurposes
TheAustralianCopyrightAct1968(theAct)allowsamaximumof
onechapteror10%ofthepagesofthispublication,whicheveristhegreater,
tobereproducedand/orcommunicatedbyanyeducationalinstitution
foritseducationalpurposesprovidedthattheeducationalinstitution
(orthebodythatadministersit)hasgivenaremunerationnoticeto
CopyrightAgencyLimited(CAL)undertheAct.
FordetailsoftheCALlicenceforeducationalinstitutionscontact:
CopyrightAgencyLimited
Level12,66GoulburnStreet
SydneyNSW2000
Telephone:[phone]
Facsimile:[phone]
Email:[email protected]
Reproductionandcommunicationforotherpurposes
ExceptaspermittedundertheAct(forexampleafairdealingforthe
purposesofstudy,research,criticismorreview)nopartofthispublication
maybereproduced,storedinaretrievalsystem,communicatedor
transmittedinanyformorbyanymeanswithoutpriorwrittenpermission.
Allinquiriesshouldbemadetothepublisherattheaddressabove.
CambridgeUniversityPresshasnoresponsibilityforthepersistenceor
accuracyofURLSforexternalorthird-partyinternetwebsitesreferredtoin
thispublication,anddoesnotguaranteethatanycontentonsuchwebsitesis,
orwillremain,accurateorappropriate.Informationregardingprices,travel
timetablesandotherfactualinformationgiveninthisworkiscorrectat
thetimeoffirstprintingbutCambridgeUniversityPressdoesnotguarantee
theaccuracyofsuchinformationthereafter.
Cambridge Senior Maths AC/VC E ISBN 978-1-107-56765-8 © Evans et al. 2016 Cambridge University Press
Specialist Mathematics 1&2 Photocopying is restricted under law and this material must not be transferred to another party. Updated May 2021
Contents
Introduction ix
Acknowledgements xi
AnoverviewoftheCambridgecompleteteacherandlearningresource xii
1
AlgebraI 1
1A Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1B Standardform . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1C Solvinglinearequationsandsimultaneouslinearequations . . . 8
1D Solvingproblemswithlinearequations . . . . . . . . . . . . . 13
1E Solvingproblemswithsimultaneouslinearequations . . . . . . 17
1F Substitutionandtranspositionofformulas . . . . . . . . . . . . 19
1G Algebraicfractions . . . . . . . . . . . . . . . . . . . . . . . . 22
1H Literalequations . . . . . . . . . . . . . . . . . . . . . . . . . 25
1I UsingaCAScalculatorforalgebra . . . . . . . . . . . . . . . . 28
ReviewofChapter1 . . . . . . . . . . . . . . . . . . . . . . . 33
2
Numbersystemsandsets 39
2A Setnotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2B Setsofnumbers. . . . . . . . . . . . . . . . . . . . . . . . . . 43
2C Themodulusfunction . . . . . . . . . . . . . . . . . . . . . . 48
2D Surds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2E Naturalnumbers . . . . . . . . . . . . . . . . . . . . . . . . . 57
2F LinearDiophantineequations . . . . . . . . . . . . . . . . . . 62
2G TheEuclideanalgorithm . . . . . . . . . . . . . . . . . . . . . 66
2H Problemsinvolvingsets . . . . . . . . . . . . . . . . . . . . . . 70
ReviewofChapter2 . . . . . . . . . . . . . . . . . . . . . . . 74
Cambridge Senior Maths AC/VC E ISBN 978-1-107-56765-8 © Evans et al. 2016 Cambridge University Press
Specialist Mathematics 1&2 Photocopying is restricted under law and this material must not be transferred to another party. Updated May 2021
iv Contents
3
Variation 82
3A Directvariation . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3B Inversevariation . . . . . . . . . . . . . . . . . . . . . . . . . 87
3C Fittingdata . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3D Jointvariation . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3E Partvariation . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
ReviewofChapter3 . . . . . . . . . . . . . . . . . . . . . . . 104
4
Sequencesandseries 110
4A Introductiontosequences . . . . . . . . . . . . . . . . . . . . 111
4B Arithmeticsequences . . . . . . . . . . . . . . . . . . . . . . . 118
4C Arithmeticseries . . . . . . . . . . . . . . . . . . . . . . . . . 122
4D Geometricsequences . . . . . . . . . . . . . . . . . . . . . . . 127
4E Geometricseries . . . . . . . . . . . . . . . . . . . . . . . . . 133
4F Zeno’sparadoxandinfinitegeometricseries. . . . . . . . . . . 137
ReviewofChapter4 . . . . . . . . . . . . . . . . . . . . . . . 140
5
AlgebraII 146
5A Polynomialidentities . . . . . . . . . . . . . . . . . . . . . . . 147
5B Quadraticequations . . . . . . . . . . . . . . . . . . . . . . . 151
5C Applyingquadraticequationstorateproblems . . . . . . . . . 157
5D Partialfractions . . . . . . . . . . . . . . . . . . . . . . . . . . 162
5E Simultaneousequations . . . . . . . . . . . . . . . . . . . . . 169
ReviewofChapter5 . . . . . . . . . . . . . . . . . . . . . . . 173
6
RevisionofChapters1–5 177
6A Technology-freequestions . . . . . . . . . . . . . . . . . . . . 177
6B Multiple-choicequestions . . . . . . . . . . . . . . . . . . . . 179
6C Extended-responsequestions . . . . . . . . . . . . . . . . . . 182
7
Principlesofcounting 190
7A Basiccountingmethods . . . . . . . . . . . . . . . . . . . . . 191
7B Factorialnotationandpermutations . . . . . . . . . . . . . . . 195
7C Permutationswithrestrictions . . . . . . . . . . . . . . . . . . 201
7D Permutationsoflikeobjects . . . . . . . . . . . . . . . . . . . 204
7E Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
7F Combinationswithrestrictions . . . . . . . . . . . . . . . . . . 212
7G Pascal’striangle . . . . . . . . . . . . . . . . . . . . . . . . . . 216
7H Thepigeonholeprinciple . . . . . . . . . . . . . . . . . . . . . 219
7I Theinclusion–exclusionprinciple . . . . . . . . . . . . . . . . . 223
ReviewofChapter7 . . . . . . . . . . . . . . . . . . . . . . . 228
Cambridge Senior Maths AC/VC E ISBN 978-1-107-56765-8 © Evans et al. 2016 Cambridge University Press
Specialist Mathematics 1&2 Photocopying is restricted under law and this material must not be transferred to another party. Updated May 2021
Contents v
8
Numberandproof 232
8A Directproof . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
8B Proofbycontrapositive . . . . . . . . . . . . . . . . . . . . . . 238
8C Proofbycontradiction . . . . . . . . . . . . . . . . . . . . . . 242
8D Equivalentstatements . . . . . . . . . . . . . . . . . . . . . . 246
8E Disprovingstatements . . . . . . . . . . . . . . . . . . . . . . 249
8F Mathematicalinduction . . . . . . . . . . . . . . . . . . . . . 251
ReviewofChapter8 . . . . . . . . . . . . . . . . . . . . . . . 260
9
Geometryintheplaneandproof 265
9A Points,linesandangles . . . . . . . . . . . . . . . . . . . . . . 266
9B Trianglesandpolygons . . . . . . . . . . . . . . . . . . . . . . 272
9C Congruenceandproofs . . . . . . . . . . . . . . . . . . . . . 277
9D Pythagoras’theorem . . . . . . . . . . . . . . . . . . . . . . . 282
9E Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
9F Anintroductiontosimilarity . . . . . . . . . . . . . . . . . . . 288
9G Proofsinvolvingsimilarity . . . . . . . . . . . . . . . . . . . . 295
9H Areas,volumesandsimilarity . . . . . . . . . . . . . . . . . . . 297
9I Thegoldenratio . . . . . . . . . . . . . . . . . . . . . . . . . 304
ReviewofChapter9 . . . . . . . . . . . . . . . . . . . . . . . 308
10
Circlegeometry 316
10A Anglepropertiesofthecircle . . . . . . . . . . . . . . . . . . . 317
10B Tangents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
10C Chordsincircles . . . . . . . . . . . . . . . . . . . . . . . . . 326
ReviewofChapter10 . . . . . . . . . . . . . . . . . . . . . . 329
11
RevisionofChapters7–10 334
11A Technology-freequestions . . . . . . . . . . . . . . . . . . . . 334
11B Multiple-choicequestions . . . . . . . . . . . . . . . . . . . . 337
11C Extended-responsequestions . . . . . . . . . . . . . . . . . . 342
12
Samplingandsamplingdistributions 347
12A Populationsandsamples . . . . . . . . . . . . . . . . . . . . . 348
12B Thedistributionofthesampleproportion . . . . . . . . . . . . 353
12C Investigatingthedistributionofthesampleproportion
usingsimulation . . . . . . . . . . . . . . . . . . . . . . . . . 366
12D Investigatingthedistributionofthesamplemean
usingsimulation . . . . . . . . . . . . . . . . . . . . . . . . . 373
ReviewofChapter12 . . . . . . . . . . . . . . . . . . . . . . 381
Cambridge Senior Maths AC/VC E ISBN 978-1-107-56765-8 © Evans et al. 2016 Cambridge University Press
Specialist Mathematics 1&2 Photocopying is restricted under law and this material must not be transferred to another party. Updated May 2021
vi Contents
13
Trigonometricratiosandapplications 387
13A Reviewingtrigonometry . . . . . . . . . . . . . . . . . . . . . 388
13B Thesinerule . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
13C Thecosinerule . . . . . . . . . . . . . . . . . . . . . . . . . . 397
13D Theareaofatriangle . . . . . . . . . . . . . . . . . . . . . . . 400
13E Circlemensuration . . . . . . . . . . . . . . . . . . . . . . . . 403
13F Anglesofelevation,anglesofdepressionandbearings . . . . . 408
13G Problemsinthreedimensions . . . . . . . . . . . . . . . . . . 412
13H Anglesbetweenplanesandmoredifficult3Dproblems . . . . . 416
ReviewofChapter13 . . . . . . . . . . . . . . . . . . . . . . 421
14
Furthertrigonometry 427
14A Symmetryproperties . . . . . . . . . . . . . . . . . . . . . . . 428
14B Thetangentfunction . . . . . . . . . . . . . . . . . . . . . . . 430
14C ReciprocalfunctionsandthePythagoreanidentity . . . . . . . 433
14D Additionformulasanddoubleangleformulas . . . . . . . . . . 438
14E Simplifyingacosx+bsinx . . . . . . . . . . . . . . . . . . . . 445
ReviewofChapter14 . . . . . . . . . . . . . . . . . . . . . . 448
15
Graphingtechniques 453
15A Reciprocalfunctions . . . . . . . . . . . . . . . . . . . . . . . 454
15B Locusofpoints . . . . . . . . . . . . . . . . . . . . . . . . . . 459
15C Parabolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462
15D Ellipses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465
15E Hyperbolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469
15F Parametricequations . . . . . . . . . . . . . . . . . . . . . . . 474
15G Polarcoordinates . . . . . . . . . . . . . . . . . . . . . . . . . 483
15H Graphingusingpolarcoordinates . . . . . . . . . . . . . . . . 485
15I Furthergraphingusingpolarcoordinates . . . . . . . . . . . . 488
ReviewofChapter15 . . . . . . . . . . . . . . . . . . . . . . 493
16
Complexnumbers 498
16A Startingtobuildthecomplexnumbers. . . . . . . . . . . . . . 499
16B Multiplicationanddivisionofcomplexnumbers . . . . . . . . . 503
16C Arganddiagrams . . . . . . . . . . . . . . . . . . . . . . . . . 509
16D Solvingequationsoverthecomplexnumbers . . . . . . . . . . 513
16E Polarformofacomplexnumber . . . . . . . . . . . . . . . . . 515
ReviewofChapter16 . . . . . . . . . . . . . . . . . . . . . . 520
Cambridge Senior Maths AC/VC E ISBN 978-1-107-56765-8 © Evans et al. 2016 Cambridge University Press
Specialist Mathematics 1&2 Photocopying is restricted under law and this material must not be transferred to another party. Updated May 2021
Contents vii
17
RevisionofChapters13–16 524
17A Technology-freequestions . . . . . . . . . . . . . . . . . . . . 524
17B Multiple-choicequestions . . . . . . . . . . . . . . . . . . . . 526
17C Extended-responsequestions . . . . . . . . . . . . . . . . . . 531
18
Matrices 535
18A Matrixnotation . . . . . . . . . . . . . . . . . . . . . . . . . . 536
18B Addition,subtractionandmultiplicationbyarealnumber . . . 540
18C Multiplicationofmatrices . . . . . . . . . . . . . . . . . . . . 544
18D Identities,inversesanddeterminantsfor2×2matrices . . . . . 547
18E Solutionofsimultaneousequationsusingmatrices . . . . . . . 552
ReviewofChapter18 . . . . . . . . . . . . . . . . . . . . . . 555
19
Transformationsoftheplane 560
19A Lineartransformations . . . . . . . . . . . . . . . . . . . . . . 561
19B Geometrictransformations . . . . . . . . . . . . . . . . . . . . 565
19C Rotationsandgeneralreflections. . . . . . . . . . . . . . . . . 571
19D Compositionoftransformations . . . . . . . . . . . . . . . . . 574
19E Inversetransformations. . . . . . . . . . . . . . . . . . . . . . 577
19F Transformationsofstraightlinesandothergraphs . . . . . . . 581
19G Areaanddeterminant . . . . . . . . . . . . . . . . . . . . . . 585
19H Generaltransformations . . . . . . . . . . . . . . . . . . . . . 590
ReviewofChapter19 . . . . . . . . . . . . . . . . . . . . . . 593
20
Vectors 598
20A Introductiontovectors . . . . . . . . . . . . . . . . . . . . . . 599
20B Componentsofvectors . . . . . . . . . . . . . . . . . . . . . . 607
20C Scalarproductofvectors . . . . . . . . . . . . . . . . . . . . . 611
20D Vectorprojections. . . . . . . . . . . . . . . . . . . . . . . . . 614
20E Geometricproofs . . . . . . . . . . . . . . . . . . . . . . . . . 618
20F Vectorsinthreedimensions . . . . . . . . . . . . . . . . . . . 621
ReviewofChapter20 . . . . . . . . . . . . . . . . . . . . . . 624
21
RevisionofChapters18–20 629
21A Technology-freequestions . . . . . . . . . . . . . . . . . . . . 629
21B Multiple-choicequestions . . . . . . . . . . . . . . . . . . . . 631
21C Extended-responsequestions . . . . . . . . . . . . . . . . . . 635
Cambridge Senior Maths AC/VC E ISBN 978-1-107-56765-8 © Evans et al. 2016 Cambridge University Press
Specialist Mathematics 1&2 Photocopying is restricted under law and this material must not be transferred to another party. Updated May 2021
viii Contents
22
Kinematics 640
22A Position,velocityandacceleration . . . . . . . . . . . . . . . . 641
22B Applicationsofantidifferentiationtokinematics . . . . . . . . . 646
22C Constantacceleration. . . . . . . . . . . . . . . . . . . . . . . 650
22D Velocity–timegraphs . . . . . . . . . . . . . . . . . . . . . . . 653
ReviewofChapter22 . . . . . . . . . . . . . . . . . . . . . . 659
23
Staticsofaparticle 666
23A Forcesandtriangleofforces . . . . . . . . . . . . . . . . . . . 667
23B Resolutionofforces. . . . . . . . . . . . . . . . . . . . . . . . 672
ReviewofChapter23 . . . . . . . . . . . . . . . . . . . . . . 676
24
RevisionofChapters22–23 679
24A Technology-freequestions . . . . . . . . . . . . . . . . . . . . 679
24B Multiple-choicequestions . . . . . . . . . . . . . . . . . . . . 681
24C Extended-responsequestions . . . . . . . . . . . . . . . . . . 683
Glossary 685
Answers 698
IncludedintheInteractiveTextbookandPDFtextbookonly
Chapter25:Statistics
25A Summarisingunivariatedata
25B Displayingbivariatedata
25C Thecorrelationcoefficient
25D Linesonscatterplots
25E Theleastsquaresregressionline
ReviewofChapter25
Chapter26:Logicandalgebra
26A Setsandcircuits
26B Booleanalgebra
26C Logicalconnectivesandtruthtables
26D Logiccircuits
26E Karnaughmaps
ReviewofChapter26
Chapter27:Graphtheory
27A Graphsandadjacencymatrices
27B EulercircuitsandHamiltoncycles
27C Matrixpowersandwalks
27D Completegraphs,bipartitegraphsandtrees
27E Euler’sformulaandthePlatonicsolids
27F Appendix:Wheneveryvertexhasevendegree
ReviewofChapter27
AppendixA:GuidetoTI-NspireCASCXwithOS4.0
AppendixB:GuidetoCasioClassPadII
Cambridge Senior Maths AC/VC E ISBN 978-1-107-56765-8 © Evans et al. 2016 Cambridge University Press
Specialist Mathematics 1&2 Photocopying is restricted under law and this material must not be transferred to another party. Updated May 2021
Introduction
CambridgeSpecialistMathematicsAustralianCurriculum/VCEUnits1&2providesa
completeteachingandlearningresourcefortheVCEStudyDesigntobeimplementedin
2016. Ithasbeenwrittenwithunderstandingasitschiefaimandwithamplepracticeoffered
throughtheworkedexamplesandexercises. Alltheworkhasbeentrialledintheclassroom,
andtheapproachesofferedarebasedonclassroomexperienceandtheresponsesofteachers
toearlierversionsofthisbook.
SpecialistMathematicsUnits1&2offersthematerialontopicsfromtheSpecialist
MathematicsStudyDesign. Thetopicscoveredprovideexcellentbackgroundforastudent
proceedingtoSpecialistMathematicsUnits3&4. Italsowouldbeveryusefulforastudent
proceedingtoMathematicalMethodsUnits3&4.
Thebookhasbeencarefullypreparedtoreflecttheprescribedcourse. Newmaterialhas
beenincludedformanyofthetopicsincludinggeometry,proof,statistics,transformations,
countingprinciplesandalgebra.
Thebookcontainsfiverevisionchapters. Theseprovidetechnology-free,multiple-choiceand
extended-responsequestions.
TheTI-NspirecalculatorexamplesandinstructionshavebeencompletedbyRussellBrown
andthosefortheCasioClassPadhavebeencompletedbyMariaSchaffner.
Areas of Study
ThechaptersinthisbookcoverthediversityoftopicsthatfeatureintheSpecialist
MathematicsStudyDesign. TheyarecollectedintoAreasofStudy. TopicsfromGeneral
MathematicsUnits1&2arealsoavailabletobeincorporatedintoaSpecialistMathematics
course.
ThetableoppositeshowshowcoursescanbeconstructedfromSpecialistMathematics
topics(indicatedbySM,withprescribedtopicsmarkedassuch)andGeneralMathematics
topics(indicatedbyGM). ‘ITBextra’referstoachapterthatisaccessedonlyinthe
InteractiveTextbook.
Cambridge Senior Maths AC/VC E ISBN 978-1-107-56765-8 © Evans et al. 2016 Cambridge University Press
Specialist Mathematics 1&2 Photocopying is restricted under law and this material must not be transferred to another party. Updated May 2021