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 y2 = 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