About this resource vii
About eBookPLUS and studyON x
Acknowledgements xi
Unit 1 Combinatorics, Vectors and Proof 1
Topic 1 Combinatorics
1 Permutations and combinations 1
1.1 Overview 1
1.2 Counting techniques 2
1.3 Factorials and permutations 14
1.4 Permutations with restrictions 21
1.5 Combinations 29
1.6 Applications of permutations and combinations 38
1.7 Pascal’s triangle and the pigeon-hole principle 45
1.8 Review: exam practice 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 Vectors and scalars 59
2.3 Position vectors in the plane 67
2.4 Scalar multiplication of vectors 77
2.5 The scalar (dot) product 81
2.6 The projection of vectors — scalar and vector resolutes 88
2.7 Review: exam practice 94
Answers 97
3 Applications of vectors in the plane 100
3.1 Overview 100
3.2 Displacement and velocity 101
3.3 Force and the triangle of forces 107
3.4 Force and the state of equilibrium 116
3.5 Relative velocity 127
3.6 Review: exam practice 131
Answers 134
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 Number systems and writing propositions 141
4.3 Direct proofs using Euclidean geometry 158
4.4 Indirect methods of proof 166
4.5 Proofs with rational and irrational numbers 170
4.6 Review: exam practice 176
Answers 179
5 Circle geometry 181
5.1 Overview 181
5.2 Review of congruent triangle tests 182
5.3 Circle properties 1 — angles in a circle and chords 184
5.4 Circle properties 2 — tangents, secants and segments 194
5.5 Circle properties 3 — cyclic quadrilaterals 202
5.6 Geometric proofs using vectors 208
5.7 Review: exam practice 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 Introduction to complex numbers 228
6.3 Basic operations using complex numbers 233
6.4 Complex conjugates and division of complex numbers 238
6.5 The complex plane (the Argand plane) 244
6.6 Complex numbers in polar form 252
6.7 Basic operations on complex numbers in polar form 263
6.8 Roots of equations 272
6.9 Review: exam practice 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 Sketching graphs of y = |f (x) | and y = f (|x|) from y = f (x) 286
7.3 Sketching graphs of reciprocal functions 295
7.4 Sketching graphs of rational functions 305
7.5 Review: exam practice 325
Answers 327
8 Trigonometric functions 345
8.1 Overview 345
8.2 Review of trigonometry 346
8.3 Solving trigonometric equations 369
8.4 The tangent function 379
8.5 The reciprocal functions 389
8.6 Modelling periodic functions 404
8.7 Review: exam practice 414
Answers 417
9 Trigonometric identities 436
9.1 Overview 436
9.2 Pythagorean identities 437
9.3 Compound angle formulas 447
9.4 Multiple angle formulas 455
9.5 Product–sum identities 465
9.6 Convert a cos(x) + b sin(x) to R cos(x ± ?) or R sin (x ± ?) 472
9.7 Review: exam practice 478
Answers 480
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, subtraction and scalar multiplication of matrices 487
10.3 Matrix multiplication 495
10.4 Determinants and inverses 502
10.5 Matrix equations and solving 2 × 2 linear equations 511
10.6 Review: exam practice 520
Answers 523
11 Matrix transformations 527
11.1 Overview 527
11.2 Translations 528
11.3 Reflections and rotations 535
11.4 Dilations 549
11.5 Combinations of transformations 555
11.6 Review: exam practice 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
