About this resource vi
About eBookPLUS and studyON ix
Acknowledgements x
1 Proof by mathematical induction 1
1.1 Overview 1
1.2 Introduction to proof by mathematical induction 2
1.3 Proof of divisibility 7
1.4 Further proof by induction 9
1.5 Review: exam practice 18
Answers 19
Revision Unit 3 Mathematical Induction, and Further Vectors, Matrices and Complex Numbers
Topic 1 Proof by mathematical induction 21
2 Vectors in three dimensions 22
2.1 Overview 22
2.2 Introduction to vectors in three dimensions 23
2.3 Geometric proofs using vectors 42
2.4 Cartesian and parametric equations 50
2.5 The vector equation of a straight line 65
2.6 The vector product 73
2.7 Applications of vectors 86
2.8 Review: exam practice 97
Answers 101
3 Solving systems of linear equations and the application of matrices 106
3.1 Overview 106
3.2 Solving linear equations using matrix algebra 107
3.3 Solving a system of linear equations using Gaussian elimination 119
3.4 The three cases for solutions of systems of linear equations 126
3.5 Using technology for matrix calculations 147
3.6 Dominance and Leslie matrices 153
3.7 Applications of matrices 162
3.8 Review: exam practice 180
Answers 184
4 Vector calculus 189
4.1 Overview 189
4.2 Position vectors as functions of time: circles, ellipses and hyperbolas 190
4.3 Differentiation of vectors 200
4.4 Integration of vectors 211
4.5 Straight line motion with constant and variable acceleration 217
4.6 Projectile motion 228
4.7 Circular motion 242
4.8 Review: exam practice 249
Answers 252
Revision Unit 3 Mathematical Induction, and Further Vectors, Matrices and Complex Numbers
Topic 2 Vectors and matrices 260
5 Complex numbers 261
5.1 Overview 261
5.2 Complex numbers in Cartesian form 262
5.3 Complex numbers in polar form 268
5.4 De Moivreâs theorem 278
5.5 The complex plane (the Argand plane) 283
5.6 Roots of complex numbers 291
5.7 Factorisation of polynomials 298
5.8 Review: exam practice 304
Answers 306
Revision Unit 3 Mathematical Induction, and Further Vectors, Matrices and Complex Numbers
Topic 3 Complex numbers 2 313
Practice Assessment 1 Specialist Mathematics: Problem solving and modelling task 314
Practice Assessment 2 Specialist Mathematics: Unit 3 examination 317
6 Integration techniques 323
6.1 Overview 323
6.2 Integration by linear substitution 324
6.3 Integration by non-linear substitutions 333
6.4 Integration using the trigonometric identities 341
6.5 Integration of inverse trigonometric functions 352
6.6 Integration by parts 374
6.7 Integration involving partial fractions 378
6.8 Review: exam practice 387
Answers 389
7 Applications of integral calculus 395
7.1 Overview 395
7.2 Area between a function and the axes 396
7.3 Area between functions 403
7.4 Volumes of solids of revolution 413
7.5 Volumes of revolution 421
7.6 Approximation using Simpsonâs rule 432
7.7 Exponential probability density function 441
7.8 Review: exam practice 446
Answers 448
Revision Unit 4 Further Calculus and Statistical Inference
Topic 1 Integration and applications of integration 451
8 Rates of change and differential equations 452
8.1 Overview 452
8.2 Implicit differentiation 453
8.3 Related rates as instances of the chain rule 460
8.4 Solving differential equations of the form dy/dx = f(x) 468
8.5 Solving differential equations of the form dy/dx= g(y) 475
8.6 Solving differential equations of the form dy/dx= f(x)g(y) 482
8.7 Review: exam practice 486
Answers 489
9 Applications of first-order differential equations 493
9.1 Overview 493
9.2 Growth and decay 494
9.3 Other applications of first-order differential equations 501
9.4 Bounded growth and Newtonâs law of cooling 508
9.5 Chemical reactions and dilution problems 514
9.6 The logistic equation 524
9.7 Slope fields 534
9.8 Review: exam practice 545
Answers 549
10 Modelling motion 1 554
10.1 Overview 554
10.2 A body in equilibrium under concurrent forces 555
10.3 Action and reaction forces 568
10.4 Momentum and resultant force 576
10.5 Forces on connected particles 587
10.6 Review: exam practice 595
Answers 599
11 Modelling motion 2 601
11.1 Overview 601
11.2 Forces that depend on time 602
11.3 Forces that depend on velocity 608
11.4 Forces that depend on displacement 620
11.5 Simple harmonic motion 626
11.6 Review: exam practice 634
Answers 637
Revision Unit 4 Further Calculus and Statistical Inference
Topic 2 Rates of change and differential equations 639
12 Statistical inference 640
12.1 Overview 640
12.2 Review of continuous random variables and the normal distribution 641
12.3 Sample means and simulations 644
12.4 Confidence intervals 652
12.5 Applications of confidence intervals 659
12.6 Review: exam practice 672
Answers 675
Revision Unit 4 Further Calculus and Statistical Inference
Topic 3 Statistical inference 677
Practice Assessment 3 Specialist Mathematics: Unit 4 examination 678
Practice Assessment 4 Specialist Mathematics: Units 3 & 4 examination 685
Glossary 694
Index 697
