Jacaranda Math Quest Unit 3 & 4

Cover
Title page
Copyright page
Contents
About this resource
eBookPLUS features
studyON — an invaluable exam preparation tool
About eBookPLUS and studyON
Acknowledgements
1 Proof by mathematical induction
	1.1 Overview
	1.2 Introduction to proof by mathematical induction
		1.2.1 Proof by induction basics
		1.2.2 Using Σ notation
		1.2.3 Review of set theory and logic symbols
	1.3 Proof of divisibility
		1.3.1 Divisibility results
		1.3.2 Using divisibility notation
	1.4 Further proof by induction
		1.4.1 Using other notations
		1.4.2 Forming a proposition and proving it by induction
		1.4.3 Pascal’s triangle
	1.5 Review: exam practice
	Answers
REVISION UNIT 3 Mathematical induction, and further vectors, matrices and complex numbers
	TOPIC 1 Proof by mathematical induction
2 Vectors in three dimensions
	2.1 Overview
	2.2 Introduction to vectors in three dimensions
		2.2.1 Review of vectors in two-dimensions
		2.2.2 Introduction to vectors in three-dimensions
		2.2.3 The dot product
		2.2.4 Review of vector projections (vector resolutes)
		2.2.5 Three dimensional vectors in polar form
	2.3 Geometric proofs using vectors
		2.3.1 Geometrical shapes
	2.4 Cartesian and parametric equations
		2.4.1 Cartesian coordinates for three-dimensional space
		2.4.2 Distances between points in three-dimensions
		2.4.3 The Cartesian equation of a sphere
		2.4.4 Parametric equations in two-dimensions
		2.4.5 Parametric equations as vector functions
	2.5 The vector equation of a straight line
		2.5.1 Lines in two dimensions
		2.5.2 The vector equation of a straight line in two dimensions
		2.5.3 Lines in three dimensions
		2.5.4 Vector equations in parametric and Cartesian form
	2.6 The vector product
		2.6.1 Multiplying vectors to produce vectors
		2.6.2 Applications: the area of parallelograms and triangles
		2.6.3 The vector product in î − ĵ − kˆ form
		2.6.4 Calculating the vector product using determinants
		2.6.5 Properties of the vector product
		2.6.6 The scalar triple product
	2.7 Applications of vectors
		2.7.1 The vector and Cartesian equation of a plane in three dimensions
		2.7.2 Torque
		2.7.3 Applications of vector functions of time
	2.8 Review: exam practice
	Answers
3 Solving systems of linear equations and the application of matrices
	3.1 Overview
	3.2 Solving linear equations using matrix algebra
		3.2.1 System of linear equations
		3.2.2 Solving systems of linear equations using inverse matrices
		3.2.3 Matrix algebra
	3.3 Solving a system of linear equations using Gaussian elimination
		3.3.1 Gaussian elimination
	3.4 The three cases for solutions of systems of linear equations
		3.4.1 Geometric interpretation of solutions of lines in two-dimensions
		3.4.2 Geometric interpretation of solutions of lines in three-dimensions
		3.4.3 Geometric interpretation of solutions of planes in three-dimensions
		3.4.4 The intersection of lines and planes
	3.5 Using technology for matrix calculations
		3.5.1 Matrix operations using a graphic calculator
		3.5.2 Powers of matrices
	3.6 Dominance and Leslie matrices
		3.6.1 Dominance matrices
		3.6.2 Leslie matrices
	3.7 Applications of matrices
		3.7.1 Markov chains and eigenvectors
		3.7.2 Eigenvalues
		3.7.3 Leontief Matrices
		3.7.4 Cryptology
	3.8 Review: exam practice
	Answers
4 Vector calculus
	4.1 Overview
	4.2 Position vectors as functions of time: circles, ellipses and hyperbolas
		4.2.1 Position vectors as functions
		4.2.2 Parametric and Cartesian equations of circles, ellipses and hyperbolas
	4.3 Differentiation of vectors
		4.3.1 Vector functions
		4.3.2 Rules for differentiating vectors
		4.3.3 Vectors describing motion
		4.3.4 Extension to three dimensions
		4.3.5 The gradient of the curve
		4.3.6 Applications of vector calculus
	4.4 Integration of vectors
		4.4.1 The constant vector
		4.4.2 Rules for integrating vectors
		4.4.3 Determining the Cartesian equation
	4.5 Straight line motion with constant and variable acceleration
		4.5.1 Rectilinear motion
		4.5.2 Motion with variable acceleration
		4.5.3 Motion under constant acceleration
	4.6 Projectile motion
		4.6.1 General theory of a projectile
		4.6.2 Proofs involving projectile motion
		4.6.3 Incorporating air resistance and three-dimensional motion
	4.7 Circular motion
		4.7.1 Uniform circular motion
		4.7.2 Angular velocity
		4.7.3 Velocity
		4.7.4 Acceleration
	4.8 Review: exam practice
	Answers
REVISION UNIT 3 Mathematical induction, and further vectors, matrices and complex numbers
	TOPIC 2 Vectors and matrices
5 Complex numbers
	5.1 Overview
	5.2 Complex numbers in cartesian form
		5.2.1 The definition of a complex number
		5.2.2 Review of basic operations on complex numbers in cartesian form
		5.2.3 The Complex conjugate of a complex number
		5.2.4 Inverses and division of complex numbers
	5.3 Complex numbers in polar form
		5.3.1 Review of complex numbers in polar form
		5.3.2 The complex conjugate in polar form
		5.3.3 Basic operations on complex numbers in polar form
		5.3.4 Proving identities involving the modulus and argument
	5.4 De Moivre’s theorem
	5.5 The complex plane (the Argand plane)
		5.5.1 Circles
		5.5.2 Lines
		5.5.3 Intersection of lines and circles
		5.5.4 Rays
	5.6 Roots of complex numbers
		5.6.1 Determine and examine the nthroots of complex numbers
		5.6.2 Determine and examine the nth roots of unity
		5.6.3 Solving zn = a where a ∈ R
	5.7 Factorisation of polynomials
		5.7.1 The fundamental theorem of algebra
		5.7.2 Solving quadratic Equations
		5.7.3 Solving cubic equations
		5.7.4 Solving quartic equations
	Answers
REVISION UNIT 3 Mathematical induction, and further vectors, matrices and complex numbers
	TOPIC 3 Complex numbers 2
PRACTICE ASSESSMENT 1 Specialist Mathematics: Problem solving and modelling task
	Conditions
	Context
	Task
	Approach to problem-solving and modelling
PRACTICE ASSESSMENT 2 Specialist Mathematics: Unit 3 examination
	Conditions
6 Integration techniques
	6.1 Overview
	6.2 Integration by linear substitution
		6.2.1 Determining integrals in the form ∫(ax + b)n dx
		6.2.2 Evaluating definite integrals using a linear substitution
		6.2.3 Other linear substitutions
		6.2.4 Definite integrals using other linear substitutions
	6.3 Integration by non-linear substitutions
		6.3.1 Indefinite integration using non-linear substitutions
		6.3.2 Definite integration using non-linear substitutions
	6.4 Integration using the trigonometric identities
		6.4.1 Review of trigonometric identities
		6.4.2 Integrating even powers of sin(ax) or cos(ax)
		6.4.3 Integrating sinn (ax) cosn (ax), n ≥ 1
		6.4.4 Integrating sin(ax) cosn (ax) or cos(ax) sinn (ax), n > 1
		6.4.5 Integrating odd powers of sin(ax) or cos(ax)
		6.4.6 Integrating products of sin(ax) and cos(bx)
		6.4.7 Integrating powers of tan(ax) or cot(ax)
	6.5 Integration of inverse trigonometric functions
		6.5.1 Inverse functions
		6.5.5 Derivatives of the inverse trigonometric functions
		6.5.6 Integration using the inverse trigonometric functions
	6.6 Integration by parts
		6.6.1 Introduction to integration by parts
	6.7 Integration involving partial fractions
		6.7.1 Integration by partial fractions
		6.7.2 Non-proper rational functions
		6.7.3 Rational functions involving non-linear factors
	6.8 Review: exam practice
	Answers
7 Applications of integral calculus
	7.1 Overview
	7.2 Area between a function and the axes
		7.2.1 The domain of the integrand
		7.2.2 Areas above the x-axis
		7.2.3 Areas below the x-axis
		7.2.4 Areas above and below the x-axis
	7.3 Area between functions
		7.3.1 Area between two functions that do not intersect in the required interval
		7.3.2 Area between two functions that intersect in the required interval
		7.3.3 Using Technology
	7.4 Volumes of solids of revolution
		7.4.1 Rotations about the x-axis
		7.4.2 Rotations about the y-axis
		7.4.3 Applications
	7.5 Volumes of revolution
		7.5.1 Rotations about the x-axis
		7.5.2 Rotations about the y-axis
		7.5.3 Composite figures
	7.6 Approximation using Simpson’s rule
		7.6.1 Simpson’s rule
	7.7 Exponential probability density function
		7.7.1 Exponential probability density function
	7.8 Review: exam practice
	Answers
REVISION UNIT 4 Further calculus and statistical inference
	TOPIC 1 Integration and applications of integration
8 Rates of change and differential equations
	8.1 Overview
	8.2 Implicit differentiation
		8.2.1 Introduction to implicit differentiation
		8.2.2 Parametric differentiation
	8.3 Related rates as instances of the chain rule
		8.3.1 Introduction
	8.4 Solving differential equations of the form dy/dx = f(x)
		8.4.1 Classification of differential equations
		8.4.2 Classifying solutions to a differential equation
		8.4.3 Differential equations of the form dy/dx = f(x)
	8.5 Solving differential equations of the form dy/dx = g(y)
		8.5.1 Invert, integrate and transpose
	8.6 Solving differential equations of the form dy/dx = f(x)g(y)
		8.6.1 Separation of variables
	8.7 Review: exam practice
	Answers
9 Applications of first-order differential equations
	9.1 Overview
	9.2 Growth and decay
		9.2.1 Introduction
		9.2.2 The law of natural growth
		9.2.3 Population growth
		9.2.4 Radioactive decay
		9.2.5 Half-lives
	9.3 Other applications of first-order differential equations
		9.3.1 Miscellaneous types
		9.3.2 Other population models
		9.3.3 Population models with regular removal
	9.4 Bounded growth and Newton’s law of cooling
		9.4.1 Bounded growth models
		9.4.2 Newton’s law of cooling
	9.5 Chemical reactions and dilution problems
		9.5.1 Input–output mixing problems
		9.5.2 Equal input and output flow rates
		9.5.3 Chemical reaction rates
	9.6 The logistic equation
		9.6.1 Introduction
		9.6.2 Logistic growth
		9.6.3 Points of inflection
		9.6.4 Analysis of the logistic solution
	9.7 Slope fields
		9.7.1 Introduction
		9.7.2 Sketching slope fields using technology
		9.7.3 Interpreting a slope field
	9.8 Review: exam practice
	Answers
10 Modelling motion 1
	10.1 Overview
	10.2 A body in equilibrium under concurrent forces
		10.2.1 Statics
		10.2.2 Some definitions
		10.2.3 Equilibrium
		10.2.4 Angles other than right angles
		10.2.5 Lami’s theorem
		10.2.6 Other types of forces
		10.2.7 Resolving forces
		10.2.8 Resolving all the forces
	10.3 Action and reaction forces
		10.3.1 Other types of forces-Action and reaction
		10.3.2 Inclined planes
		10.3.3 Connected particles
	10.4 Momentum and resultant force
		10.4.1 Newton’s laws of motion
		10.4.2 Momentum
		10.4.3 Constant acceleration formulas
		10.4.4 Newton’s third law of motion
		10.4.5 Motion on inclined planes
		10.4.6 Resolution of all the forces
	10.5 Forces on connected particles
		10.5.1 Two or more particles
		10.5.2 Particles connected by smooth pulleys
		10.5.3 Atwood’s machine
		10.5.4 Connected vehicles
	10.6 Review: exam practice
	Answers
11 Modelling motion 2
	11.1 Overview
	11.2 Forces that depend on time
		11.2.1 Setting up the equation of motion
		11.2.2 Integrals involving trigonometric functions
		11.2.3 Horizontal rectilinear motion
	11.3 Forces that depend on velocity
		11.3.1 Setting up the equation of motion
		11.3.2 Horizontal rectilinear motion
		11.3.3 General cases
		11.3.4 Vertical motion
	11.4 Forces that depend on displacement
		11.4.1 Setting up the equation of motion
		11.4.2 Relationships between time, displacement, velocity and acceleration
		11.4.3 Expressing x in terms of t
	11.5 Simple harmonic motion
		11.5.1 Definition of simple harmonic motion
	11.6 Review: exam practice
	Answers
REVISION UNIT 4 Further calculus and statistical inference
	TOPIC 2 Rates of change and differential equations
12 Statistical inference
	12.1 Overview
	12.2 Review of continuous random variables and the normal distribution
		12.2.1 Continuous random variables
		12.2.2 The normal distribution
		12.2.3 Linear combinations of random variables
	12.3 Sample means and simulations
		12.3.1 Estimating population parameters
		12.3.2 The distribution of sample means
		12.3.3 Verifying the formulae
	12.4 Confidence intervals
		12.4.1 Confidence intervals
		12.4.2 Calculation of confidence intervals
		12.4.3 Verifying the formulae
	12.5 Applications of confidence intervals
		12.5.1 Using confidence intervals to estimate other confidence intervals
		12.5.2 Simulating sampling
		12.5.3 Confirming normality
	12.6 Review: exam practice
	Answers
REVISION UNIT 4 Further calculus and statistical inference
	TOPIC 3 Statistical inference
PRACTICE ASSESSMENT 3 Specialist Mathematics: Unit 4 examination
	Conditions
PRACTICE ASSESSMENT 4 Specialist Mathematics: Units 3 & 4 examination
	Conditions
Glossary
Index