About this resource vii

About eBookPLUS and studyON x

Acknowledgements xi

**1 Lines and linear relationships 1**

1.1 Overview 1

1.2 Linearly related variables, linear equations and inequations 3

1.3 Systems of 3 × 3 simultaneous linear equations 16

1.4 Linear graphs and their equations 21

1.5 Intersections of lines and their applications 34

1.6 Coordinate geometry of the straight line 40

1.7 Bisection and lengths of line segments 47

1.8 Review: exam practice 53

Answers 56

**2 Algebraic foundations 63**

2.1 Overview 63

2.2 Algebraic skills 65

2.3 Pascal’s triangle and binomial expansions 73

2.4 The binomial theorem 77

2.5 Sets of real numbers 85

2.6 Surds 91

2.7 Review: exam practice 102

Answers 106

**3 Quadratic relationships 110**

3.1 Overview 110

3.2 Quadratic equations with rational roots 112

3.3 Quadratics over *R* 117

3.4 Applications of quadratic equations 130

3.5 Graphs of quadratic polynomials 134

3.6 Determining the rule of a quadratic polynomial from a graph 146

3.7 Quadratic inequations 152

3.8 Quadratic models and applications 159

3.9 Review: exam practice 163

Answers 167

**4 Cubic polynomials 178**

4.1 Overview 178

4.2 Polynomials 180

4.3 The remainder and factor theorems 192

4.4 Graphs of cubic polynomials 201

4.5 Equations of cubic polynomials 212

4.6 Cubic models and applications 223

4.7 Review: exam practice 228

Answers 232

**5 Higher-degree polynomials 247**

5.1 Overview 247

5.2 Quartic polynomials 249

5.3 Families of polynomials 258

5.4 Numerical approximations to roots of polynomial equations 267

5.5 Review: exam practice 276

Answers 280

**6 Functions and relations 289**

6.1 Overview 289

6.2 Functions and relations 291

6.3 The circle 301

6.4 The rectangular hyperbola and the truncus 312

6.5 The relation *y*2 = *x* 330

6.6 Other functions and relations 343

6.7 Transformations of functions 356

6.8 Review: exam practice 366

Answers 371

**Revision Topics 1 to 6 393**

**7 Matrices and applications to transformations 394**

7.1 Overview 394

7.2 Addition, subtraction and scalar multiplication of matrices 396

7.3 Matrix multiplication 403

7.4 Determinants and inverses of 2 × 2 matrices 408

7.5 Matrix equations and solving 2 × 2 linear simultaneous equations 414

7.6 Translations 424

7.7 Reflections 431

7.8 Dilations 438

7.9 Combinations of transformations 443

7.10 Review: exam practice 446

Answers 451

**Revision Topic 7 458**

**8 Probability 459**

8.1 Overview 459

8.2 Probability review 461

8.3 Conditional probability 472

8.4 Independence 481

8.5 Counting techniques 487

8.6 Binomial coefficients and Pascal’s triangle 500

8.7 Review: exam practice 509

Answers 513

**Revision Topic 8 517**

**9 Trigonometric functions 1 518**

9.1 Overview 518

9.2 Trigonometric ratios 519

9.3 Circular measure 529

9.4 Unit circle definitions 538

9.5 Symmetry properties 548

9.6 Graphs of the sine and cosine functions 559

9.7 Review: exam practice 570

Answers 573

**10 Trigonometric functions 2 580**

10.1 Overview 580

10.2 Trigonometric equations 582

10.3 Transformations of sine and cosine graphs 591

10.4 Applications of sine and cosine functions 605

10.5 The tangent function 612

10.6 Trigonometric relationships 622

10.7 Review: exam practice 629

Answers 634

**11 Exponential functions 648**

11.1 Overview 648

11.2 Indices as exponents 650

11.3 Indices as logarithms 658

11.4 Graphs of exponential functions 668

11.5 Applications of exponential functions 677

11.6 Inverses of exponential functions 684

11.7 Review: exam practice 697

Answers 701

**Revision Topics 9 to 11 712**

**12 Introduction to differential calculus 713**

12.1 Overview 713

12.2 Rates of change 715

12.3 Gradients of secants 723

12.4 The derivative function 728

12.5 Differentiation of polynomials by rule 735

12.6 Review: exam practice 746

Answers 750

**13 Differentiation and applications 757**

13.1 Overview 757

13.2 Limits, continuity and differentiability 759

13.3 Derivatives of power functions 769

13.4 Coordinate geometry applications of differentiation 777

13.5 Curve sketching 786

13.6 Optimisation problems 796

13.7 Rates of change and kinematics 803

13.8 Review: exam practice 812

Answers 815

**14 Anti-differentiation and introduction to integral calculus 824**

14.1 Overview 824

14.2 Anti-derivatives 826

14.3 Anti-derivative functions and graphs 833

14.4 Application of anti-differentiation 841

14.5 The definite integral 847

14.6 Review: exam practice 858

Answers 862

**Revision Topics 12 to 14 868**

Glossary 869

Index 878