About This Resource vii

Acknowledgements xiv

1 Combinatorics 1

1.1 Overview 2

1.2 Counting techniques 3

1.3 Factorials and permutations 15

1.4 Permutations with restrictions 23

1.5 Combinations 31

1.6 Applications of permutations and combinations 40

1.7 Pascal’s triangle and the pigeon-hole principle 47

1.8 Review 55

Answers 58

2 Sequences and series 63

2.1 Overview 64

2.2 Describing sequences 65

2.3 Arithmetic sequences 69

2.4 Arithmetic series 75

2.5 Geometric sequences 78

2.6 Geometric series 86

2.7 Applications of sequences and series 92

2.8 Review 101

Answers 105

3 Logic and algorithms 111

3.1 Overview 112

3.2 Statements (propositions), connectives and truth tables 113

3.3 Valid and invalid arguments 120

3.4 Boolean algebra and digital logic 131

3.5 Sets and Boolean algebra 139

3.6 Algorithms and pseudocode 149

3.7 Review 166

Answers 170

4 Proof and number 189

4.1 Overview 190

4.2 Number systems and mathematical statements 191

4.3 Direct and indirect methods of proof 206

4.4 Proofs with rational and irrational numbers 212

4.5 Proof by mathematical induction 218

4.6 Proof of divisibility using induction 223

4.7 Review 227

Answers 230

5 Matrices 233

5.1 Overview 234

5.2 Addition, subtraction and scalar multiplication of matrices 235

5.3 Matrix multiplication 243

5.4 Determinants and inverses 251

5.5 Matrix equations 260

5.6 Review 271

Answers 275

6 Graph theory 279

6.1 Overview 280

6.2 Introduction to graph theory 281

6.3 Planar graphs and Euler’s formula 300

6.4 Eulerian and Hamiltonian graphs 311

6.5 Weighted graphs and trees 321

6.6 Bipartite graphs and the Hungarian algorithm 331

6.7 Review 343

Answers 353

7 Trigonometric ratios and applications 367

7.1 Overview 368

7.2 Review of trigonometry 369

7.3 The sine rule 379

7.4 The cosine rule 385

7.5 Arc length, sectors and segments 390

7.6 Review 398

Answers 402

8 Trigonometric identities 405

8.1 Overview 406

8.2 Pythagorean identities 407

8.3 Compound angle formulas 416

8.4 Double and half angle formulas 426

8.5 Converting a cos(x) + b sin(x) to r cos(x ± α) or r sin(x ± α) 435

8.6 Review 441

Answers 444

9 Vectors in the plane 449

9.1 Overview 450

9.2 Vectors and scalars 451

9.3 Position vectors in the plane 459

9.4 Scalar multiplication of vectors 470

9.5 The scalar (dot) product 475

9.6 Projections of vectors—scalar and vector resolutes 483

9.7 Review 489

Answers 492

10 Complex numbers 497

10.1 Overview 498

10.2 Introduction to complex numbers 499

10.3 Basic operations on complex numbers 504

10.4 Complex conjugates and division of complex numbers 510

10.5 The complex plane (the Argand plane) 517

10.6 Complex numbers in polar form 524

10.7 Basic operations on complex numbers in polar form 536

10.8 Solving quadratic equations with complex roots 546

10.9 Lines, rays, circles, ellipses and regions in the complex plane 550

10.10 Review 562

Answers 565

11 Transformations 577

11.1 Overview 578

11.2 Translations 579

11.3 Reflections and rotations 588

11.4 Dilations 603

11.5 Combinations of transformations 610

11.6 Review 619

Answers 623

12 Functions, relations and graphs 627

12.1 Overview 628

12.2 The absolute value function 629

12.3 Partial fractions 636

12.4 Reciprocal graphs 645

12.5 The reciprocal trigonometric functions 654

12.6 Inverse trigonometric functions 669

12.7 Circles and ellipses 683

12.8 Hyperbolas 695

12.9 Polar coordinates, equations and graphs 703

12.10 Parametric equations 712

12.11 Review 717

Answers 721

13 Simulation, sampling and sampling distributions 745

13.1 Overview 746

13.2 Random experiments, events and event spaces 747

13.3 Simulation 750

13.4 Discrete random variables 757

13.5 Sampling 768

13.6 Sampling distribution of sample means 774

13.7 Review 786

Answers 791

Glossary 795

Index 803