Jacaranda Maths Quest Units 1&2 Specialist Mathematics 11 for Queensland

610 Pages·2018·17.414 MB·English

by  Catherine Smith, Nick Simpson, Raymond Rozen, Pauline Holland, Paula Evans, Steven Morris, Margarent Swale

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JACARANDA MATHS QUEST
UNITS
SPECIALIST 11
1& 2
MATHEMATICS
FOR QUEENSLAND
CATHERINE SMITH | NICK SIMPSON | RAYMOND ROZEN
CONTRIBUTING AUTHORS
Pauline Holland | Paula Evans | Steven Morris | Margaret Swale
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Firstpublished2018by
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Firstpublished2018by
JohnWiley&SonsAustralia,Ltd
42McDougallStreet,Milton,Qld4064
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ISBN:978-0-7303-5720-9
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CONTENTS
Aboutthisresource................................................................................................................................................................................................. vii
AbouteBookPLUSandstudyON....................................................................................................................................................................... x
Acknowledgements................................................................................................................................................................................................. xi
UNIT 1 COMBINATORICS, VECTORS AND PROOF 1
TOPIC 1 Combinatorics
1 Permutations and combinations 1
1.1 Overview...................................................................................................................................................................................... 1
1.2 Countingtechniques............................................................................................................................................................... 2
1.3 Factorialsandpermutations................................................................................................................................................ 14
1.4 Permutationswithrestrictions............................................................................................................................................. 21
1.5 Combinations............................................................................................................................................................................. 29
1.6 Applicationsofpermutationsandcombinations......................................................................................................... 38
1.7 Pascal’striangleandthepigeon-holeprinciple........................................................................................................... 45
1.8 Review:exampractice........................................................................................................................................................... 52
Answers................................................................................................................................................................................................... 54
REVISION UNIT 1 Combinatorics, vectors and proof
TOPIC 1 Combinatorics
........................................................................................................... 57
TOPIC 2 Vectors in the plane
2 Vectors in the plane 58
2.1 Overview...................................................................................................................................................................................... 58
2.2 Vectorsandscalars................................................................................................................................................................. 59
2.3 Positionvectorsintheplane............................................................................................................................................... 67
2.4 Scalarmultiplicationofvectors.......................................................................................................................................... 77
2.5 Thescalar(dot)product......................................................................................................................................................... 81
2.6 Theprojectionofvectors—scalarandvectorresolutes........................................................................................ 88
2.7 Review:exampractice........................................................................................................................................................... 94
Answers................................................................................................................................................................................................... 97
3 Applications of vectors in the plane 100
3.1 Overview......................................................................................................................................................................................100
3.2 Displacementandvelocity....................................................................................................................................................101
3.3 Forceandthetriangleofforces..........................................................................................................................................107
3.4 Forceandthestateofequilibrium.....................................................................................................................................116
3.5 Relativevelocity........................................................................................................................................................................127
3.6 Review:exampractice...........................................................................................................................................................131
Answers...................................................................................................................................................................................................134
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REVISION UNIT 1 Combinatorics, vectors and proof
TOPIC 2 Vectors in the plane
................................................................................................136
PRACTICE ASSESSMENT 1
Problem solving and modelling task 137
..............................................................................
TOPIC 3 Introduction to proof
4 Introduction to proof 140
4.1 Overview......................................................................................................................................................................................140
4.2 Numbersystemsandwritingpropositions....................................................................................................................141
4.3 DirectproofsusingEuclideangeometry.........................................................................................................................158
4.4 Indirectmethodsofproof......................................................................................................................................................166
4.5 Proofswithrationalandirrationalnumbers...................................................................................................................170
4.6 Review:exampractice...........................................................................................................................................................176
Answers...................................................................................................................................................................................................179
5 Circle geometry 181
5.1 Overview......................................................................................................................................................................................181
5.2 Reviewofcongruenttriangletests....................................................................................................................................182
5.3 Circleproperties1—anglesinacircleandchords...................................................................................................184
5.4 Circleproperties2—tangents,secantsandsegments..........................................................................................194
5.5 Circleproperties3—cyclicquadrilaterals....................................................................................................................202
5.6 Geometricproofsusingvectors.........................................................................................................................................208
5.7 Review:exampractice...........................................................................................................................................................217
Answers...................................................................................................................................................................................................220
REVISION UNIT 1 Combinatorics, vectors and proof
TOPIC 3 Introduction to proof
..............................................................................................221
PRACTICE ASSESSMENT 2
Unit 1 examination 222
...................................................................................................................
UNIT 2 COMPLEX NUMBERS, TRIGONOMETRY, FUNCTIONS
AND MATRICES 227
TOPIC 1 Complex numbers 1
6 Complex numbers 227
6.1 Overview......................................................................................................................................................................................227
6.2 Introductiontocomplexnumbers.....................................................................................................................................228
iv CONTENTS
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6.3 Basicoperationsusingcomplexnumbers.....................................................................................................................233
6.4 Complexconjugatesanddivisionofcomplexnumbers..........................................................................................238
6.5 Thecomplexplane(theArgandplane)............................................................................................................................244
6.6 Complexnumbersinpolarform.........................................................................................................................................252
6.7 Basicoperationsoncomplexnumbersinpolarform................................................................................................263
6.8 Rootsofequations...................................................................................................................................................................272
6.9 Review:exampractice...........................................................................................................................................................276
Answers...................................................................................................................................................................................................278
REVISION UNIT 2 Complex numbers, trigonometry, functions
and matrices
TOPIC 1 Complex numbers 1
...............................................................................................284
TOPIC 2 Trigonometry and functions
7 Sketching graphs 285
7.1 Overview......................................................................................................................................................................................285
7.2 Sketchinggraphsofy=|f(x)|andy=f(|x|)fromy=f(x)...................................................................................286
7.3 Sketchinggraphsofreciprocalfunctions.......................................................................................................................295
7.4 Sketchinggraphsofrationalfunctions............................................................................................................................305
7.5 Review:exampractice...........................................................................................................................................................325
Answers...................................................................................................................................................................................................327
8 Trigonometric functions 345
8.1 Overview......................................................................................................................................................................................345
8.2 Reviewoftrigonometry..........................................................................................................................................................346
8.3 Solvingtrigonometricequations........................................................................................................................................369
8.4 Thetangentfunction...............................................................................................................................................................379
8.5 Thereciprocalfunctions........................................................................................................................................................389
8.6 Modellingperiodicfunctions...............................................................................................................................................404
8.7 Review:exampractice...........................................................................................................................................................414
Answers...................................................................................................................................................................................................417
9 Trigonometric identities 436
9.1 Overview......................................................................................................................................................................................436
9.2 Pythagoreanidentities............................................................................................................................................................437
9.3 Compoundangleformulas...................................................................................................................................................447
9.4 Multipleangleformulas..........................................................................................................................................................455
9.5 Product–sumidentities..........................................................................................................................................................465
9.6 Convertacos(x)+b sin(x)toR cos(x±𝛼)orR sin(x±𝛼)..................................................................................472
9.7 Review:exampractice...........................................................................................................................................................478
Answers...................................................................................................................................................................................................480
CONTENTS v
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REVISION UNIT 2 Complex numbers, trigonometry, functions
and matrices
TOPIC 2 Trigonometry and functions
................................................................................485
TOPIC 3 Matrices
10 Matrix arithmetic 486
10.1 Overview......................................................................................................................................................................................486
10.2 Addition,subtractionandscalarmultiplicationofmatrices....................................................................................487
10.3 Matrixmultiplication................................................................................................................................................................495
10.4 Determinantsandinverses...................................................................................................................................................502
10.5 Matrixequationsandsolving2×2linearequations..................................................................................................511
10.6 Review:exampractice...........................................................................................................................................................520
Answers.....................................................................................................................................................................................................523
11 Matrix transformations 527
11.1 Overview......................................................................................................................................................................................527
11.2 Translations.................................................................................................................................................................................528
11.3 Reflectionsandrotations......................................................................................................................................................535
11.4 Dilations........................................................................................................................................................................................549
11.5 Combinationsoftransformations......................................................................................................................................555
11.6 Review:exampractice...........................................................................................................................................................564
Answers.....................................................................................................................................................................................................567
REVISION UNIT 2 Complex numbers, trigonometry, functions
and matrices
TOPIC 3 Matrices
........................................................................................................................570
PRACTICE ASSESSMENT 3
Unit 2 examination 571
...................................................................................................................
PRACTICE ASSESSMENT 4
Units 1 & 2 examination 575
........................................................................................................
Glossary...................................................................................................................................................................................................................... 583
Index............................................................................................................................................................................................................................. 593
vi CONTENTS
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ABOUT THIS RESOURCE
JacarandaMathsQuest11SpecialistMathematicsUnits1&2forQueenslandisexpertlytailoredtocomprehensivelyaddressthe
intentandstructureofthenewsyllabus.TheJacarandaMathsQuestforQueenslandseriesprovideseasy-to-followtextandis
supportedbyabankofresourcesforbothteachersandstudents.AtJacarandawebelievethateverystudentshouldexperience
successandbuildconfidence,whilethosewhowanttobechallengedaresupportedastheyprogresstomoredifficultconceptsand
questions.
Preparing students for exam success
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study, revision and exam practice
Units1&2 Area4 Sequence1 Concept5
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SkillSHEETS to support skill development Exercise6.6Complexnumbersinpolarform
and SpreadSHEETS to explore Technologyfree
mathematical relationships and concepts Ienx1p.thrbaees..fsoRCeldealoplacwrsueialsnaegctneoqtmtuzheme=seoti4xnoan+mcst8ugdilitivoispetnalaenarncogefA(o𝜋𝜋zr)f.gozarnfrAdordmgia(tgzhr)eacmoorr.irgeicnt.t(oD3odneoctimusaelypolaucrecsawlchuelaretotrh.)eanglecannotbeeasily
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CHAPTER6Complexnumbers261
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In the Prelims section of (cid:31) (cid:31)
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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.
viii ABOUT THIS RESOURCE
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studyON — an invaluable exam preparation tool
studyON provides a complete study solution. An interactive and highly visual online study, revision
and exam practice tool, it is designed to help students and teachers maximise exam results.
Concept summary screens Direct links from the The studyON question hierarchy allows students in the
and interactivities summarise eBookPLUS help scaffold Continue Studying feature to revise across the entire
key concepts and help prevent students’ understanding and course, or to drill down to concept level for a more granular
misconceptions. study practices. set of questions.
studyON prepares students for actual exams: studyON’s built‑in progress tracker enables
• The Sit Exams feature allows students to sit timed practice exams. self‑diagnosis of strengths and weaknesses at
• Exam‑style questions have been authored by our team of highly qualiḀed teachers. a topic and concept level, so students know
• From 2020, offcial past QCAA exam questions will be available for Units 3 & 4 with exemplary exactly what needs extra revision and can sit
worked solutions to provide feedback for every question.
their exams with confidence.
studyON Teacher edition is a powerful diagnostic tool
Enables teachers to assign Allows teachers to monitor students’ Alignment with the Jacaranda text helps
activities for extra revision and activities and results to pinpoint with planning, and instant feedback saves
practice, and track progress strengths and weaknesses. Armed with marking time. Built‑in reporting functionality
at an individual, group evidence‑based insights, teachers can lets teachers easily schedule, export and print
and classroom level intervene at the right time. reports in Excel.
ABOUT THIS RESOURCE ix
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“FMPrelims_print” — 2018/9/16 — 7:39 — page xii — #10
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About eBookPLUS and studyON
Access your online Jacaranda
STEP 1 Go to www.jacplus.com.au
resources anywhere, anytime, from and create a user account.
STEP 2 Enter your registration code.
any device in three easy steps: STEP 3 Instant access!
eBookPLUS is an electronic version of the studyON is an interactive and highly visual online
textbook, together with a targeted range of study, revision and exam practice tool designed to help
supporting multimedia resources. students and teachers maximise exam results.
eBookPLUS features:  studyON features:
• eBook — the entire textbook in • Concept summary screens provide concise
electronic format explanations of key concepts, with relevant examples.
• Digital documents designed for easy • Access exam questions that have been written
customisation and editing by experienced examiners for practice at a concept,
area or entire course level, and receive immediate
• Interactivities to reinforce and enhance
feedback. From 2020, QCAA questions will be
students’ learning
included with exemplary worked solutions.
• eLessons — engaging video clips and
• Sit past QCAA exams (Units 3 & 4) or topic tests
supporting material
(Units 1 & 2) in exam-like situations.
• Weblinks to relevant support material on
• Video animations and interactivities
the internet
demonstrate concepts to provide a deep
eGuidePLUS features assessment and understanding (Units 3 & 4 only).
curriculum material to support teachers.
• All results and performance in practice and
sit questions are tracked to a concept level to
pinpoint strengths and weaknesses.
NEED HELP? Go to www.jacplus.com.au Minimum requirements
and select the Help link.
JacarandaPLUS requires you to use a
• Visit the JacarandaPLUS Support Centre at supported internet browser and version,
http://jacplus.desk.com to access a range of step-by-step otherwise you will not be able to access
user guides, ask questions or search for information. your resources or view all features and
• Contact John Wiley & Sons Australia, Ltd. upgrades. Please view the complete list of
Email: [email protected] JacPLUS minimum system requirements at
Phone: 1800 JAC PLUS (1800 522 7587) http://jacplus.desk.com.
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