SPECIALIST
MATHEMATICS
VCE UNITS 1 & 2
CAMBRIDGE SENIOR MATHEMATICS VCE
SECOND EDITION
DAVID TREEBY | MICHAEL EVANS | DOUGLAS WALLACE
GARETH AINSWORTH | KAY LIPSON
,SPECIALIST
MATHEMATICS
VCE UNITS 1 & 2
CAMBRIDGE SENIOR MATHEMATICS VCE
SECOND EDITION
DAVID TREEBY | MICHAEL EVANS | DOUGLAS WALLACE
GARETH AINSWORTH | KAY LIPSON
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,Contents
Introduction and overview ix
Acknowledgements xiv
1 Reviewing algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . .
1A Indices
1
2
1B Standard form . . . . . . . . . . . . . . . . . . . . . . . . 5
1C Solving linear equations and simultaneous linear equations . . 8
1D Solving problems with linear equations . . . . . . . . . . . . 13
1E Solving problems with simultaneous linear equations . . . . . 17
1F Substitution and transposition of formulas . . . . . . . . . . 19
1G Algebraic fractions . . . . . . . . . . . . . . . . . . . . . . 22
1H Literal equations . . . . . . . . . . . . . . . . . . . . . . . 25
1I Using a CAS calculator for algebra . . . . . . . . . . . . . . 28
Review of Chapter 1 . . . . . . . . . . . . . . . . . . . . . . 32
2 Number systems and.sets
2A . . . . . . . . . . . . . . . . . . . . . . . . .
Set notation
38
39
2B Sets of numbers . . . . . . . . . . . . . . . . . . . . . . . 42
2C Surds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2D Natural numbers . . . . . . . . . . . . . . . . . . . . . . . 52
2E Problems involving sets . . . . . . . . . . . . . . . . . . . . 57
Review of Chapter 2 . . . . . . . . . . . . . . . . . . . . . . 61
,iv Contents
3 Sequences and series
3A Introduction to sequences . . . . . . . . . . . . . . . . . .
67
68
3B Arithmetic sequences . . . . . . . . . . . . . . . . . . . . . 75
3C Arithmetic series . . . . . . . . . . . . . . . . . . . . . . . 79
3D Geometric sequences . . . . . . . . . . . . . . . . . . . . . 85
3E Geometric series . . . . . . . . . . . . . . . . . . . . . . . 90
3F Applications of geometric sequences . . . . . . . . . . . . . 93
3G Recurrence relations of the form tn = rtn−1 + d . . . . . . . . 100
3H Zeno’s paradox and infinite geometric series . . . . . . . . . 109
Review of Chapter 3 . . . . . . . . . . . . . . . . . . . . . . 112
4 Additional algebra
4A . . . . . . . . . . . . . . .
Polynomial identities . . . . . .
119
120
4B Quadratic equations . . . . . . . . . . . . . . . . . . . . . 124
4C Applying quadratic equations to rate problems . . . . . . . . 130
4D Partial fractions . . . . . . . . . . . . . . . . . . . . . . . 135
4E Simultaneous equations . . . . . . . . . . . . . . . . . . . 142
Review of Chapter 4 . . . . . . . . . . . . . . . . . . . . . . 146
5 Revision of Chapters 1–4
5A Technology-free questions . . . . . . . . . . . . . . . . . . 152
152
5B Multiple-choice questions . . . . . . . . . . . . . . . . . . . 154
5C Extended-response questions . . . . . . . . . . . . . . . . . 157
5D Investigations . . . . . . . . . . . . . . . . . . . . . . . . . 165
6 Proof 6A Direct proof . . . . . . . . . . . . . . . . . . . . . . . . . . 169
168
6B Proof by contrapositive . . . . . . . . . . . . . . . . . . . . 174
6C Proof by contradiction. . . . . . . . . . . . . . . . . . . . 178
6D Equivalent statements . . . . . . . . . . . . . . . . . . . . 182
6E Disproving statements . . . . . . . . . . . . . . . . . . . . 185
6F Mathematical induction . . . . . . . . . . . . . . . . . . . . 189
Review of Chapter 6 . . . . . . . . . . . . . . . . . . . . . . 198
7 Logic 7A The algebra of sets . . . . . . . . . . . . . . . . . . . . . . 204
203
7B Switching circuits . . . . . . . . . . . . . . . . . . . . . . . 210
7C Boolean algebra . . . . . . . . . . . . . . . . . . . . . . . 214
7D Logical connectives and truth tables . . . . . . . . . . . . . 221
, Contents v
7E Valid arguments . . . . . . . . . . . . . . . . . . . . . . . 230
7F Logic circuits . . . . . . . . . . . . . . . . . . . . . . . . . 235
7G Karnaugh maps . . . . . . . . . . . . . . . . . . . . . . . . 239
Review of Chapter 7 . . . . . . . . . . . . . . . . . . . . . . 245
8 Algorithms
8A Introduction to algorithms . . . . . . . . . . . . . . . . . .
250
251
8B Iteration and selection . . . . . . . . . . . . . . . . . . . . 257
8C Introduction to pseudocode . . . . . . . . . . . . . . . . . . 263
8D Further pseudocode . . . . . . . . . . . . . . . . . . . . . 270
Review of Chapter 8 . . . . . . . . . . . . . . . . . . . . . . 279
9 Combinatorics
9A Basic counting methods . . . . . . . . . . . . . . . . . . . . 289
288
9B Factorial notation and permutations . . . . . . . . . . . . . 293
9C Permutations with restrictions . . . . . . . . . . . . . . . . 299
9D Permutations of like objects . . . . . . . . . . . . . . . . . . 302
9E Combinations . . . . . . . . . . . . . . . . . . . . . . . . . 304
9F Combinations with restrictions . . . . . . . . . . . . . . . . 309
9G . . . . . . . . .
Pascal’s triangle . . . . . . . . . . . . . . 313
9H The pigeonhole principle. . . . . . . . . . . . . . . . . . . 316
9I The inclusion–exclusion principle . . . . . . . . . . . . . . . 320
Review of Chapter 9 . . . . . . . . . . . . . . . . . . . . . . 325
10 Revision of Chapters 6–9
10A Technology-free questions . . . . . . . . . . . . . . . . . .
329
329
10B Multiple-choice questions . . . . . . . . . . . . . . . . . . . 333
10C Extended-response questions . . . . . . . . . . . . . . . . . 338
10D Investigations . . . . . . . . . . . . . . . . . . . . . . . . . 343
11 Matrices
11A Matrix notation . . . . . . . . . . . . . . . . . . . . . . . . 346
345
11B Addition, subtraction and multiplication by a real number . . . 350
11C Multiplication of matrices . . . . . . . . . . . . . . . . . . . 354
11D . . . 357
Identities, inverses and determinants for 2 × 2 matrices
11E Solution of simultaneous equations using matrices . . . . . . 362
11F Inverses and determinants for n × n matrices . . . . . . . . . 365
11G Simultaneous linear equations with more than two variables . 372
Review of Chapter 11 . . . . . . . . . . . . . . . . . . . . . 377
,vi Contents
12 Graph theory
12A Graphs and adjacency matrices . . . . . . . . . . . . . . . . 383
382
12B Euler circuits . . . . . . . . . . . . . . . . . . . . . . . . . 391
12C Hamiltonian cycles . . . . . . . . . . . . . . . . . . . . . . 398
12D Using matrix powers to count walks in graphs . . . . . . . . 402
12E Regular, cycle, complete and bipartite graphs . . . . . . . . . 405
12F Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410
12G Euler’s formula and the Platonic solids . . . . . . . . . . . . 414
12H Appendix: When every vertex has even degree . . . . . . . . 421
Review of Chapter 12 . . . . . . . . . . . . . . . . . . . . . 423
13 Revision of Chapters 11–12
13A Technology-free questions . . . . . . . . . . . . . . . . . .
429
429
13B Multiple-choice questions . . . . . . . . . . . . . . . . . . . 431
13C Extended-response questions . . . . . . . . . . . . . . . . . 434
13D Investigations . . . . . . . . . . . . . . . . . . . . . . . . . 436
14 Simulation, sampling and sampling distributions
14A Expected value and variance for discrete random variables . . 440
439
14B Distribution of sums of random variables . . . . . . . . . . . 447
14C Populations and samples . . . . . . . . . . . . . . . . . . . 456
14D Investigating the distribution of the sample mean
using simulation . . . . . . . . . . . . . . . . . . . . . . . 461
Review of Chapter 14 . . . . . . . . . . . . . . . . . . . . . 471
15 Trigonometric ratios and applications
15A . . . . . . . . . . . . . . . . . . . .
Reviewing trigonometry
477
478
15B The sine rule . . . . . . . . . . . . . . . . . . . . . . . . . 483
15C The cosine rule . . . . . . . . . . . . . . . . . . . . . . . . 487
15D The area of a triangle . . . . . . . . . . . . . . . . . . . . . 490
15E Circle mensuration . . . . . . . . . . . . . . . . . . . . . . 493
15F Angles of elevation, angles of depression and bearings . . . . 498
15G Problems in three dimensions . . . . . . . . . . . . . . . . . 502
15H Angles between planes and more difficult 3D problems . . . . 506
Review of Chapter 15 . . . . . . . . . . . . . . . . . . . . . 511
16 Trigonometric identities
16A Reciprocal circular functions and the Pythagorean identity . . 518
517
16B Compound and double angle formulas . . . . . . . . . . . . 523
16C Simplifying a cos x + b sin x . . . . . . . . . . . . . . . . . . 530
16D Sums and products of sines and cosines . . . . . . . . . . . 533
Review of Chapter 16 . . . . . . . . . . . . . . . . . . . . . 537
, Contents vii
17 Graphing functions and relations
17A The inverse circular functions . . . . . . . . . . . . . . . .
542
543
17B Reciprocal functions . . . . . . . . . . . . . . . . . . . . . 549
17C Graphing the reciprocal circular functions . . . . . . . . . . 553
17D The modulus function . . . . . . . . . . . . . . . . . . . . . 557
17E Locus of points . . . . . . . . . . . . . . . . . . . . . . . . 564
17F Parabolas . . . . . . . . . . . . . . . . . . . . . . . . . . . 568
17G Ellipses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571
17H Hyperbolas . . . . . . . . . . . . . . . . . . . . . . . . . . 575
17I Parametric equations . . . . . . . . . . . . . . . . . . . . . 580
17J Polar coordinates . . . . . . . . . . . . . . . . . . . . . . . 589
17K Graphing using polar coordinates . . . . . . . . . . . . . . . 591
Review of Chapter 17 . . . . . . . . . . . . . . . . . . . . . 597
18 Complex numbers
18A Starting to build the complex numbers . . . . . . . . . . . .
604
605
18B Multiplication and division of complex numbers . . . . . . . . 609
18C Argand diagrams . . . . . . . . . . . . . . . . . . . . . . . 616
18D Solving quadratic equations over the complex numbers . . . . 620
18E Solving polynomial equations over the complex numbers . . . 623
18F Polar form of a complex number . . . . . . . . . . . . . . . 626
18G Sketching subsets of the complex plane . . . . . . . . . . . . 631
Review of Chapter 18 . . . . . . . . . . . . . . . . . . . . . 638
19 Revision of Chapters 15–18
19A Technology-free questions . . . . . . . . . . . . . . . . . .
643
643
19B Multiple-choice questions . . . . . . . . . . . . . . . . . . . 645
19C Extended-response questions . . . . . . . . . . . . . . . . . 650
19D Investigations . . . . . . . . . . . . . . . . . . . . . . . . . 654
20 Transformations of the plane
20A Linear transformations . . . . . . . . . . . . . . . . . . . . 657
656
20B Geometric transformations . . . . . . . . . . . . . . . . . . 661
20C Rotations and general reflections . . . . . . . . . . . . . . . 667
20D Composition of transformations . . . . . . . . . . . . . . . . 670
20E Inverse transformations. . . . . . . . . . . . . . . . . . . 673
20F Transformations of straight lines and other graphs . . . . . . 677
20G Area and determinant . . . . . . . . . . . . . . . . . . . . . 681
20H General transformations . . . . . . . . . . . . . . . . . . . 686
Review of Chapter 20 . . . . . . . . . . . . . . . . . . . . . 689
,viii Contents
21 Vectors in the plane
21A Introduction to vectors . . . . . . . . . . . . . . . . . . . . 695
694
21B Components of vectors . . . . . . . . . . . . . . . . . . . . 703
21C Scalar product of vectors . . . . . . . . . . . . . . . . . . . 707
21D Vector projections . . . . . . . . . . . . . . . . . . . . . . 710
21E Geometric proofs . . . . . . . . . . . . . . . . . . . . . . . 714
21F Applications of vectors: displacement and velocity . . . . . . 717
21G Applications of vectors: relative velocity . . . . . . . . . . . . 722
21H Applications of vectors: forces and equilibrium . . . . . . . . 727
21I Vectors in three dimensions . . . . . . . . . . . . . . . . . . 735
Review of Chapter 21 . . . . . . . . . . . . . . . . . . . . . 738
22 Revision of Chapters 20–21
22A Technology-free questions . . . . . . . . . . . . . . . . . . 745
745
22B Multiple-choice questions . . . . . . . . . . . . . . . . . . . 747
22C Extended-response questions . . . . . . . . . . . . . . . . . 750
Glossary 753
Answers 767
Online chapter accessed through the Interactive Textbook or PDF Textbook
23 Kinematics
23A Position, velocity and acceleration
23B Applications of antidifferentiation to kinematics
23C Constant acceleration
23D Velocity–time graphs
Review of Chapter 23
Online appendices accessed through the Interactive Textbook or PDF Textbook
Appendix A Guide to the TI-Nspire CAS calculator in VCE mathematics
Appendix B Guide to the Casio ClassPad II CAS calculator in VCE mathematics
Appendix C Introduction to coding using Python
Appendix D Introduction to coding using the TI-Nspire
Appendix E Introduction to coding using the Casio ClassPad
, Introduction and
overview
Cambridge Specialist Mathematics VCE Units 1&2 Second Edition provides a complete
teaching and learning resource for the VCE Study Design to be first implemented in 2023.
It has been written with understanding as its chief aim, and with ample practice offered
through the worked examples and exercises. The work has been trialled in the classroom, and
the approaches offered are based on classroom experience and the responses of teachers to
earlier editions of this book and the requirements of the new Study Design.
The course is designed as preparation for Specialist Mathematics Units 3 and 4.
Specialist Mathematics Units 1 and 2 provide an introductory study of topics in proof,
logic, sequences, algorithms and pseudocode, graph theory, algebra, functions, statistics,
complex numbers, and vectors and their applications in a variety of practical and theoretical
contexts. Techniques of proof are discussed in Chapter 6 and the concepts discussed there are
employed in the following chapters and in Specialist Mathematics Units 3 and 4.
Chapter 1 provides an opportunity for students to revise and strengthen their algebra; and this
is revisited in Chapter 3, where polynomial identities and partial fractions are introduced.
We have also written online appendices to support teachers and students to better develop
their programming capabilities using both the programming language Python and the
inbuilt capabilities of students’ CAS calculators. Additional material on kinematics has
also been placed in an online appendix.
Five extensive revision chapters are placed at key stages throughout the book. These provide
technology-free multiple-choice and extended-response questions.
The first four revision chapters contain material suitable for student investigations, a feature
of the new course. The Study Design suggests that ‘[a]n Investigation comprises one to two
weeks of investigation into one or two practical or theoretical contexts or scenarios based
on content from areas of study and application of key knowledge and key skills for the
outcomes’. We have aimed to provide strong support for teachers in the development of these
investigations.
The TI-Nspire calculator examples and instructions have been completed by Peter Flynn,
and those for the Casio ClassPad by Mark Jelinek, and we thank them for their helpful
contributions.
