Cambridge Seniour Mathematics VCE: Specialist Mathematics Units 1 & 2

766 Pages·2015·42.885 MB·English

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

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Cambridge Senior Maths AC/VCE 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.
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/VCE 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.
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/VCE 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.
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/VCE 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.
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/VCE 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.
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/VCE 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.
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
AppendixA:GuidetotheTI-NspireCASCalculator(OS4)inVCEMathematics
AppendixB:GuidetotheCasioClassPadIICASCalculatorinVCEMathematics
Cambridge Senior Maths AC/VCE 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.
Introduction
Specialist(cid:1)Mathematics(cid:1)Australian(cid:1)Curriculum/VCE(cid:1)Units(cid:1)1(cid:1)&(cid:1)2(cid:1)provides(cid:1)a(cid:1)complete(cid:1)teaching(cid:1)
and(cid:1)learning(cid:1)resource(cid:1)for(cid:1)the(cid:1)VCE(cid:1)Study(cid:1)Design(cid:1)to(cid:1)be(cid:1)implemented(cid:1)in(cid:1)2016.(cid:1)It(cid:1)has(cid:1)been(cid:1)
written(cid:1)with(cid:1)understanding(cid:1)as(cid:1)its(cid:1)chief(cid:1)aim(cid:1)and(cid:1)with(cid:1)ample(cid:1)practice(cid:1)o↵ered(cid:1)through(cid:1)the(cid:1)
worked(cid:1)examples(cid:1)and(cid:1)exercises.(cid:1)All(cid:1)the(cid:1)work(cid:1)has(cid:1)been(cid:1)trialled(cid:1)in(cid:1)the(cid:1)classroom,(cid:1)and(cid:1)the(cid:1)
approaches(cid:1)o↵ered(cid:1)are(cid:1)based(cid:1)on(cid:1)classroom(cid:1)experience(cid:1)and(cid:1)the(cid:1)responses(cid:1)of(cid:1)teachers(cid:1)to(cid:1)earlier(cid:1)
versions(cid:1)of(cid:1)this(cid:1)book.
Specialist(cid:1)Mathematics(cid:1)Units(cid:1)1(cid:1)&(cid:1)2(cid:1)o↵ers(cid:1)the(cid:1)material(cid:1)on(cid:1)topics(cid:1)from(cid:1)the(cid:1)Specialist(cid:1)
Mathematics(cid:1)Study(cid:1)Design.(cid:1)The(cid:1)topics(cid:1)covered(cid:1)provide(cid:1)excellent(cid:1)background(cid:1)for(cid:1)a(cid:1)student(cid:1)
proceeding(cid:1)to(cid:1)Specialist(cid:1)Mathematics(cid:1)Units(cid:1)3(cid:1)&(cid:1)4.(cid:1)It(cid:1)also(cid:1)would(cid:1)be(cid:1)very(cid:1)useful(cid:1)for(cid:1)a(cid:1)student(cid:1)
proceeding(cid:1)to(cid:1)Mathematical(cid:1)Methods(cid:1)Units(cid:1)3(cid:1)&(cid:1)4.
The(cid:1)book(cid:1)has(cid:1)been(cid:1)carefully(cid:1)prepared(cid:1)to(cid:1)reflect(cid:1)the(cid:1)prescribed(cid:1)course.(cid:1)New(cid:1)material(cid:1)has(cid:1)
been(cid:1)included(cid:1)for(cid:1)many(cid:1)of(cid:1)the(cid:1)topics(cid:1)including(cid:1)geometry,(cid:1)proof,(cid:1)statistics,(cid:1)transformations,(cid:1)
counting(cid:1)principles(cid:1)and(cid:1)algebra.
The(cid:1)book(cid:1)contains(cid:1)five(cid:1)revision(cid:1)chapters.(cid:1)These(cid:1)provide(cid:1)technology-free,(cid:1)multiple-choice(cid:1)and(cid:1)
extended-response(cid:1)questions.
The(cid:1)TI-Nspire(cid:1)calculator(cid:1)examples(cid:1)and(cid:1)instructions(cid:1)have(cid:1)been(cid:1)completed(cid:1)by(cid:1)Russell(cid:1)Brown(cid:1)
and(cid:1)those(cid:1)for(cid:1)the(cid:1)Casio(cid:1)ClassPad(cid:1)have(cid:1)been(cid:1)completed(cid:1)by(cid:1)Maria(cid:1)Scha↵ner.
Areas(cid:1)of(cid:1)Study
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(cid:46)(cid:66)(cid:85)(cid:73)(cid:70)(cid:78)(cid:66)(cid:85)(cid:74)(cid:68)(cid:84)(cid:1)(cid:54)(cid:79)(cid:74)(cid:85)(cid:84)(cid:1)(cid:18)(cid:1)(cid:7)(cid:1)(cid:19)(cid:1)(cid:52)(cid:85)(cid:86)(cid:69)(cid:90)(cid:1)(cid:37)(cid:70)(cid:84)(cid:74)(cid:72)(cid:79)(cid:1)(cid:68)(cid:66)(cid:79)(cid:1)also(cid:1)(cid:67)(cid:70)(cid:1)incorporated(cid:1)into(cid:1)a(cid:1)Specialist(cid:1)
Mathematics(cid:1)course.
The(cid:1)table(cid:1)shows(cid:1)how(cid:1)courses(cid:1)can(cid:1)be(cid:1)constructed(cid:1)from(cid:1)Specialist(cid:1)Mathematics(cid:1)topics(cid:1)
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topics(cid:1)(indicated(cid:1)by(cid:1)GM).(cid:1)‘ITB(cid:1)extra’(cid:1)refers(cid:1)to(cid:1)a(cid:1)chapter(cid:1)that(cid:1)is(cid:1)accessed(cid:1)only(cid:1)in(cid:1)the
Interactive(cid:1)Textbook.
Cambridge Senior Maths AC/VCE ISBN 978-1-107-56765-8 © Evans et al. 2016 Cambridge University Press
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