Jacaranda Maths Quest Units 1&2 Mathematical Methods 11 for Queensland

768 Pages·2018·20.924 MB·English

by  Kahni Burrows, Sue Michell, Miles Ford, Renee Gordon, Shirley Sharpley, Matthew Mack, Libby Kempton, Steven Morris, Raymond Rozen, Margaret Swale

JACARANDA MATHS QUEST
UNITS
MATHEMATICAL 11
1& 2
METHODS
FOR QUEENSLAND
KAHNI BURROWS | SUE MICHELL | MILES FORD
CONTRIBUTING AUTHORS
Renée Gordon | Shirley Sharpley | Matthew Mack | Libby Kempton
Steven Morris | Raymond Rozen | Margaret Swale
Firstpublished2018by
JohnWiley&SonsAustralia,Ltd
42McDougallStreet,Milton,Qld4064
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ISBN:978-0-7303-5711-7
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10 9 8 7 6 5 4 3 2 1
CONTENTS
Aboutthisresource................................................................................................................................................................................................. ix
AbouteBookPLUSandstudyON....................................................................................................................................................................... xii
Acknowledgements................................................................................................................................................................................................. xiii
UNIT 1 ALGEBRA, STATISTICS AND FUNCTIONS 1
TOPIC 1 Arithmetic and geometric sequences and series 1
1 Arithmetic sequences 1
1.1 Overview...................................................................................................................................................................................... 1
1.2 Arithmeticsequences............................................................................................................................................................. 2
1.3 Thegeneralformofanarithmeticsequence................................................................................................................. 8
1.4 Thesumofanarithmeticsequence.................................................................................................................................. 15
1.5 Applicationsofarithmeticsequences.............................................................................................................................. 19
1.6 Review:exampractice........................................................................................................................................................... 26
Answers................................................................................................................................................................................................... 29
REVISION UNIT 1 TOPIC 1
Chapter 1
......................................................................................................................................... 31
TOPIC 2 Functions and graphs
2 Functions 32
2.1 Overview...................................................................................................................................................................................... 32
2.2 Functionsandrelations.......................................................................................................................................................... 33
2.3 Functionnotation...................................................................................................................................................................... 44
2.4 Transformationsoffunctions............................................................................................................................................... 53
2.5 Piece-wisefunctions............................................................................................................................................................... 62
2.6 Review:exampractice........................................................................................................................................................... 74
Answers................................................................................................................................................................................................... 80
3 Quadratic relationships 89
3.1 Overview...................................................................................................................................................................................... 89
3.2 Graphsofquadraticfunctions............................................................................................................................................ 90
3.3 Solvingquadraticequationswithrationalroots...........................................................................................................103
3.4 FactorisingandsolvingquadraticsoverR.....................................................................................................................109
3.5 Thediscriminant.......................................................................................................................................................................116
3.6 Modellingwithquadraticfunctions...................................................................................................................................126
3.7 Review:exampractice...........................................................................................................................................................134
Answers...................................................................................................................................................................................................138
4 Inverse proportions and graphs of relations 145
4.1 Overview......................................................................................................................................................................................145
4.2 Thehyperbola............................................................................................................................................................................146
4.3 Inverseproportion....................................................................................................................................................................157
4.4 Thecircle.....................................................................................................................................................................................162
4.5 Thesidewaysparabola..........................................................................................................................................................170
4.6 Review:exampractice...........................................................................................................................................................178
Answers...................................................................................................................................................................................................182
5 Powers and polynomials 191
5.1 Overview......................................................................................................................................................................................191
5.2 Polynomials................................................................................................................................................................................192
5.3 Graphsofcubicpolynomials...............................................................................................................................................203
5.4 Thefactorandremaindertheorems.................................................................................................................................217
5.5 Solvingcubicequations........................................................................................................................................................224
5.6 Cubicmodelsandapplications..........................................................................................................................................229
5.7 Graphsofquarticpolynomials............................................................................................................................................235
5.8 Solvingpolynomialequations.............................................................................................................................................242
5.9 Review:exampractice...........................................................................................................................................................250
Answers...................................................................................................................................................................................................254
REVISION UNIT 1 TOPIC 2
Chapters 2 to 5
.............................................................................................................................270
PRACTICE ASSESSMENT 1
Problem solving and modelling task: Functions and graphs
................................271
TOPIC 3 Counting and probability
6 Counting and probability 274
6.1 Overview......................................................................................................................................................................................274
6.2 Fundamentalsofprobability................................................................................................................................................275
6.3 Relativefrequency...................................................................................................................................................................286
6.4 Conditionalprobability...........................................................................................................................................................290
6.5 Independence............................................................................................................................................................................299
6.6 Permutationsandcombinations........................................................................................................................................305
6.7 Pascal’striangleandbinomialexpansions....................................................................................................................319
6.8 Review:exampractice...........................................................................................................................................................329
Answers...................................................................................................................................................................................................332
REVISION UNIT 1 TOPIC 3
Chapter 6
.........................................................................................................................................336
iv CONTENTS
TOPIC 4 Exponential functions 1
7 Indices 337
7.1 Overview......................................................................................................................................................................................337
7.2 Indexlaws...................................................................................................................................................................................338
7.3 Negativeandrationalindices..............................................................................................................................................343
7.4 Indicialequationsandscientificnotation.......................................................................................................................348
7.5 Review:exampractice...........................................................................................................................................................354
Answers...................................................................................................................................................................................................356
REVISION UNIT 1 TOPIC 4
Chapter 7
.........................................................................................................................................358
TOPIC 5 Arithmetic and geometric sequences and series 2
8 Geometric sequences 359
8.1 Overview......................................................................................................................................................................................359
8.2 Recursivedefinitionandthegeneraltermofgeometricsequences...................................................................360
8.3 Thesumofageometricsequence....................................................................................................................................371
8.4 Geometricsequencesincontext.......................................................................................................................................378
8.5 Review:exampractice...........................................................................................................................................................384
Answers...................................................................................................................................................................................................387
REVISION UNIT 1 TOPIC 5
Chapter 8
.........................................................................................................................................389
PRACTICE ASSESSMENT 2
Unit 1 internal examination
.....................................................................................................390
UNIT 2 CALCULUS AND FURTHER FUNCTIONS 399
TOPIC 1 The logarithmic function 1
TOPIC 2 Exponential functions 2
9 Exponential and logarithmic functions 399
9.1 Overview......................................................................................................................................................................................399
9.2 Exponentialfunctions.............................................................................................................................................................400
9.3 Logarithmicfunctions.............................................................................................................................................................410
9.4 Modellingwithexponentialfunctions...............................................................................................................................423
9.5 Solvingequationswithindices...........................................................................................................................................432
9.6 Review:exampractice...........................................................................................................................................................438
Answers...................................................................................................................................................................................................442
CONTENTS v
REVISION UNIT 2 TOPIC 1 AND 2
Chapter 9
.........................................................................................................................................449
TOPIC 3 Trigonometric functions 1
10 Trigonometric functions 450
10.1 Overview......................................................................................................................................................................................450
10.2 Trigonometryreview................................................................................................................................................................451
10.3 Radianmeasure........................................................................................................................................................................459
10.4 Unitcircledefinitions...............................................................................................................................................................469
10.5 Exactvaluesandsymmetryproperties...........................................................................................................................480
10.6 Graphsofthesine,cosineandtangentfunctions......................................................................................................490
10.7 Transformationsofsineandcosinegraphs...................................................................................................................499
10.8 Solvingtrigonometricequations........................................................................................................................................513
10.9 Modellingwithtrigonometricfunctions...........................................................................................................................524
10.10 Review:exampractice...........................................................................................................................................................531
Answers...................................................................................................................................................................................................536
REVISION UNIT 2 TOPIC 3
Chapter 10
.....................................................................................................................................550
TOPIC 4 Introduction to differential calculus
11 Rates of change 551
11.1 Overview......................................................................................................................................................................................551
11.2 Exploringratesofchange.....................................................................................................................................................552
11.3 Thedifferencequotient..........................................................................................................................................................561
11.4 Differentiatingsimplefunctions..........................................................................................................................................567
11.5 Interpretingthederivative.....................................................................................................................................................570
11.6 Review:exampractice...........................................................................................................................................................576
Answers...................................................................................................................................................................................................580
12 Properties and applications of derivatives 584
12.1 Overview......................................................................................................................................................................................584
12.2 Differentiationbyformula......................................................................................................................................................585
12.3 Thederivativeasafunction.................................................................................................................................................589
12.4 Propertiesofthederivative...................................................................................................................................................595
12.5 Differentiationofpowerandpolynomialfunctions.....................................................................................................599
12.6 Review:exampractice...........................................................................................................................................................604
Answers....................................................................................................................................................................................................607
vi CONTENTS
13 Applications of derivatives 611
13.1 Overview......................................................................................................................................................................................611
13.2 Gradientandequationofatangent..................................................................................................................................612
13.3 Displacement–timegraphs...................................................................................................................................................620
13.4 Sketchingcurvesusingderivatives...................................................................................................................................629
13.5 Modellingoptimisationproblems.......................................................................................................................................643
13.6 Review:exampractice...........................................................................................................................................................648
Answers....................................................................................................................................................................................................651
REVISION UNIT 2 TOPIC 4
Chapters 11 to 13
........................................................................................................................656
TOPIC 5 Further differentiation and applications 1
14 Differentiation rules 657
14.1 Overview......................................................................................................................................................................................657
14.2 Theproductrule........................................................................................................................................................................658
14.3 Thequotientrule.......................................................................................................................................................................660
14.4 Thechainrule.............................................................................................................................................................................666
14.5 Applicationsoftheproduct,quotientandchainrules..............................................................................................672
14.6 Review:exampractice...........................................................................................................................................................674
Answers....................................................................................................................................................................................................677
REVISION UNIT 2 TOPIC 5
Chapter 14
......................................................................................................................................682
TOPIC 6 Discrete random variables 1
15 Discrete random variables 1 683
15.1 Overview......................................................................................................................................................................................683
15.2 Discreterandomvariables....................................................................................................................................................684
15.3 Expectedvalues........................................................................................................................................................................695
15.4 Varianceandstandarddeviation........................................................................................................................................704
15.5 Applicationsofdiscreterandomvariables.....................................................................................................................712
15.6 Review:exampractice...........................................................................................................................................................718
Answers....................................................................................................................................................................................................723
REVISION UNIT 2 TOPIC 6
Chapter 15
......................................................................................................................................727
CONTENTS vii
PRACTICE ASSESSMENT 3
Unit 2 internal examination
.....................................................................................................728
PRACTICE ASSESSMENT 4
Units 1 and 2 internal examination
.....................................................................................735
Glossary...................................................................................................................................................................................................................... 743
Index............................................................................................................................................................................................................................. 750
viii CONTENTS
ABOUT THIS RESOURCE
JacarandaMathsQuest11MathematicalMethodsUnits1&2forQueenslandisexpertlytailoredtoaddress
comprehensively the intent and structure of the new syllabus. The Jacaranda Maths Quest for Queensland
series provides easy-to-follow text and is supported by a bank of resources for both teachers and students.
AtJacarandawebelievethateverystudentshouldexperiencesuccessandbuildconfidence,whilethosewho
wanttobechallengedaresupportedastheyprogresstomoredifficultconceptsandquestions.
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misconceptions
In the Prelims section of
your eBooKPLUS
A downloadable PDF of the entire
solutions manual, containing worked
solutions for every question in the text
A set of four practice assessments: a
problem solving and modelling task and
three examination-style assessments
FREE copies of the Maths Quest Manual
for the TI-Nspire CAS calculator and
the Maths Quest Manual for the Casio
Classpad II calculator A downloadable PDF of the entire print text
Additional resources
for teachers available
in the eGuidePLUS
In the Resources tab of every chapter
there are two topic tests in downloadable,
customisable Word format with
worked solutions.
In the Prelims section
of the eGuidePLUS
Work programs are provided to assist with
classroom planning.
Practice assessments: in addition to
the four provided in the eBookPLUS,
teachers have access to a further four
quarantined assessments. Modelled on
QCAA guidelines, the problem solving
and modelling tasks are provided
with exemplary responses while the
examination-style assessments include
annotated worked solutions. They are
downloadable in Word format to allow
teachers to customise as they need.
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