Jacaranda Maths Quest 12 Specialist Mathematics Units 3 and 4 for Queensland, 2e learnON & Print

Learning with learnON ix
Teaching with learnON xv
Acknowledgements xvi

Problem-solving and modelling task guide

Unit 3 Further Complex Numbers, Proof, Vectors and Matrices 1

Topic 1 Further Complex Numbers

1 Further complex numbers 3

2 Mathematical induction and trigonometric proofs 69

2.1 Overview 70
2.2 Proof by induction 71
2.3 Proofs of divisibility 85
2.4 Trigonometric proofs using De Moivre’s theorem 88
2.5 Review 96
Answers 99

TOPIC 3 VECTORS IN Two AND Three DIMENSIONS

3 Vectors in two and three dimensions 101

3.1 Overview 102
3.2 Vectors in three dimensions 104
3.3 Algebra of three dimensional vectors 120
3.4 Geometric proofs using vectors 130
3.5 Vector, parametric and Cartesian equations 142
3.6 The vector (cross) product 164
3.7 Applications of vectors 179
3.8 Review 194
Answers 198

Topic 4 Vector Calculus

4 Vector calculus 203

4.1 Overview 204
4.2 Position vectors as functions of time — circles, ellipses and hyperbolas 205
4.3 Collision problems 221
4.4 Differentiation and integration of vectors 231
4.5 Straight line motion with constant and variable acceleration 247
4.6 Applications of vector calculus 258
4.6 Review 278
Answers 283

Topic 5 Further Matrices

5 Further matrices 291

5.1 Overview 292
5.2 Solving linear equations using matrix algebra 294
5.3 Solving a system of linear equations using Gaussian elimination 315
5.4 The three cases for solutions of systems of linear equations 323
5.5 Dominance and Leslie matrices 347
5.6 Applications of matrices 357
5.7 Review 377
Answers 383

Practice Assessment 1
Problem-solving and Modelling Task

Practice Assessment 2
Unit 3 Examination

Unit 4 Further Calculus and Statistical Inference 389

Topic 1 Integration Techniques

6 Integration techniques 391

6.1 Overview 392
6.2 Integration by linear substitution 394
6.3 Integration by non-linear substitutions 406
6.4 Integration using the trigonometric identities 415
6.5 Integration of inverse trigonometric functions 428
6.6 Integration by parts 450
6.7 Integration involving partial fractions 457
6.8 Review 466
Answers 470

Topic 2 Applications of Integral Calculus

7 Applications of integral calculus 477

7.1 Overview 478
7.2 Area between a function and the axes 480
7.3 Area between functions 489
7.4 Volumes of solids of revolution 502
7.5 Volumes of revolution 512
7.6 Approximation using Simpson’s rule 524
7.7 Exponential probability density function 535
7.8 Review 542
Answers 546

Topic 3 Rates of Change and Differential Equations

8 Rates of change and differential equations 551

8.1 Overview 552
8.2 Implicit differentiation 554
8.3 Related rates as instances of the chain rule 563
8.4 Solving differential equations of the form dy/dx = f(x)
8.5 Solving differential equations of the form dy/dx = g(y)
8.6 Solving differential equations of the form dy/dx = f(x)g(y)
8.7 Slope fields 594
8.8 Review 608
Answers 614

9 Applications of first-order differential equations 621

9.1 Overview 622
9.2 Growth and decay 623
9.3 Other applications of first-order differential equations 634
9.4 Bounded growth and Newton’s law of cooling 641
9.5 Chemical reactions and dilution problems 648
9.6 The logistic equation 659
9.7 Review 672
Answers 678

Topic 4 Modelling Motion

10 Modelling motion 683

10.1 Overview 684
10.2 Equilibrium situations under concurrent forces 686
10.3 Non-equilibrium situations under concurrent forces 705
10.4 Forces that depend on time 714
10.5 Forces that depend on velocity 726
10.6 Forces that depend on displacement 744
10.7 Review 760
Answers 764

Topic 5 Statistical Inference

11 Statistical inference 767

11.1 Overview 768
11.2 Sampling distributions 770
11.3 Confidence intervals 785
11.4 Margin of error, level of confidence and sample size 795
11.5 Applications and other distributions 801
11.6 Review 817
Answers 823

Practice Assessment 3
Unit 4 Examination

Practice Assessment 4
Units 3 & 4 Examination

Glossary (online only)