Course map
Discrete mathematics
- LessonPractice
Logic and proof
Unit 3: Discrete mathematics- Write implications, converses, contrapositives and equivalences, and decide what is necessary or sufficient
- Prove statements directly, by cases, by contradiction and by contrapositive
- Use quantifiers, examples and counter-examples
- LessonPractice
Mathematical induction
Unit 4: Discrete mathematics- Prove divisibility results, inequalities, sums and products by induction
- Write the base case, the inductive step and the conclusion in full
- Find a flawed step in an attempted proof
Functions, relations and graphs
- LessonPractice
Rational and quotient functions
Unit 3: Functions, relations and graphs- Express rational functions of low degree as partial fractions
- Find asymptotes, intercepts, stationary points and points of inflection of rational and quotient functions
- Read the features of a graph from its rule, and a rule from its features
Algebra, number and structure
- LessonPractice
Complex numbers in polar form
Unit 3: Algebra, number and structure- Move between Cartesian and polar form and do arithmetic in either
- Use De Moivre's theorem for powers and roots, including roots of unity
- Describe and sketch subsets of the complex plane
- LessonPractice
Polynomials over the complex numbers
Unit 3: Algebra, number and structure- Factorise polynomials over C, using the conjugate root theorem
- Solve polynomial equations over C by completing the square and by factors
- Apply the fundamental theorem of algebra
Calculus
- LessonPractice
Differential calculus
Unit 3: Calculus- Differentiate inverse circular functions
- Use second derivatives for concavity and points of inflection
- Differentiate implicitly and solve related rates problems
- LessonPractice
Techniques of integration
Unit 3: Calculus- Anti-differentiate by recognition, substitution and trigonometric identities
- Anti-differentiate with partial fractions and integrate by parts
- Relate the graph of a function to the graphs of its anti-derivatives, and integrate numerically
- LessonPractice
Applications of integration
Unit 3: Calculus- Find areas of regions bounded by curves
- Find arc lengths of parametric curves
- Find volumes and surface areas of solids of revolution about either axis
- LessonPractice
Differential equations
Unit 4: Calculus- Formulate differential equations from rates, including the logistic equation
- Verify solutions and read direction fields
- Solve by anti-differentiation and separation of variables, and approximate with Euler's method
- LessonPractice
Kinematics
Unit 4: Calculus- Analyse rectilinear motion from velocity-time graphs
- Find displacement, velocity and acceleration with calculus
- Use the forms of acceleration, including v dv/dx, with differential equations
Space and measurement
- LessonPractice
Vectors
Unit 3: Space and measurement- Do vector arithmetic and find magnitudes, unit vectors and angles
- Use scalar and vector products, resolutes and linear dependence
- Prove geometric results with vectors
- LessonPractice
Lines, planes and curves
Unit 3: Space and measurement- Write vector and parametric equations of curves and find their Cartesian equations
- Write vector, parametric and Cartesian equations of lines
- Write vector, parametric and Cartesian equations of planes, using the vector product for a normal
- LessonPractice
Vector calculus
Unit 4: Space and measurement- Find the path traced by a position vector, including circles, ellipses and hyperbolas
- Decide whether two particles' paths cross and whether the particles meet
- Differentiate and anti-differentiate vector functions for velocity, speed and acceleration
Data analysis, probability and statistics
- LessonPractice
Linear combinations and the sample mean
Unit 4: Data analysis, probability and statistics- Find means and variances of linear combinations of random variables
- Use the normal distribution of a linear combination of independent normal variables
- Find the distribution of a sample mean
- LessonPractice
Confidence intervals for a mean
Unit 4: Data analysis, probability and statistics- Find approximate confidence intervals for a population mean
- Find the sample size or level that gives a stated interval
- Interpret confidence intervals and how they vary between samples
- LessonPractice
Hypothesis tests for a mean
Unit 4: Data analysis, probability and statistics- Write null and alternative hypotheses and find a test statistic and p-value
- Decide at a level of significance with one-tail and two-tail tests
- Interpret a result in context and relate errors to conditional probability
The examinations
VCE Specialist Mathematics Units 3 and 4. Both examinations cover Unit 3 and Unit 4. School-assessed Coursework is set by your school, so it is described in a lesson rather than practised here.