QCAA Specialist Mathematics study plan

Derive the product, quotient, and chain rules from first principles and apply each to composite algebraic and trigonometric expressions

Construct epsilon-style and counterexample proofs to justify derivative and limit claims

Solve separable first-order differential equations and verify solutions by back-substitution

Set up differential-equation models for rates of change such as cooling, growth, and mixing problems

Write a one-page summary sheet of standard derivatives and integration results without consulting notes

Apply integration techniques including substitution, by parts, and partial fractions to definite and indefinite integrals

Compute areas, volumes of revolution, and arc lengths using definite integrals

Solve linear first-order differential equations using an integrating factor

Analyse slope fields qualitatively to predict solution behaviour

Prove statements by mathematical induction, including inequality and divisibility cases

Perform arithmetic on complex numbers in Cartesian and polar form, including multiplication, division, and powers

Apply De Moivre's theorem to find roots and powers and locate them on the Argand diagram

Sketch loci defined by modulus and argument equations and inequalities and identify their geometric meaning

Derive and apply the double-angle, compound-angle, and sum-to-product identities

Solve trigonometric equations over a specified domain, expressing general solutions correctly

Convert between trigonometric and exponential forms of complex numbers and use them to simplify products and quotients

Solve trigonometric equations involving multiple angles and quadratic forms in sine or cosine

Apply identities to integrate products of trigonometric functions

Interpret complex-number transformations geometrically as rotations and dilations

Prove trigonometric identities step by step with justification for each line

Perform vector addition, scalar multiplication, dot products, and cross products in two and three dimensions

Resolve vectors into components and apply them to force, velocity, and displacement problems

Determine equations of lines and planes in vector form and find intersections and angles

Carry out matrix operations, compute determinants and inverses, and solve linear systems using row reduction

Interpret matrix transformations geometrically and identify invariant points and lines

Apply eigenvalues and eigenvectors to diagonalise small matrices and interpret the result

Model networks and transitions using matrix methods such as adjacency and transition matrices

Compute and interpret binomial and normal probabilities, including continuity corrections

Calculate confidence intervals for a mean and interpret the confidence level correctly

Apply counting principles and the pigeonhole principle to discrete problems

Translate unfamiliar real-world scenarios into mathematical models and state assumptions explicitly

Select and justify appropriate techniques from calculus, vectors, or statistics for open-ended problems

Evaluate model limitations and communicate conclusions with appropriate precision and units

Analyse discrete optimisation problems such as shortest paths and scheduling using systematic methods

Interpret statistical output critically, distinguishing correlation from causation

Synthesise techniques across all domains in multi-step problems requiring proof, calculus, and modelling

Complete full-length practice papers within strict time limits

Diagnose and repair recurring errors identified in earlier error logs

Refine written communication so that reasoning and justification are explicit and complete

Manage exam timing by allocating fixed minutes per mark and practising question triage