Jacaranda Maths Quest 11 Specialist Mathematics Units 1&2 for Queensland eBookPLUS & Print + StudyOn Specialist Mathematics U1&2 for Qld (Book Code)

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 ± 𝛼) or R sin (x ± 𝛼) 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