Mathematics Specialised Level 4 study plan

Classify and factor polynomials of degree up to four, including sum and difference of cubes and grouping patterns.

Solve rational, radical, and absolute-value equations and check for extraneous roots.

Sketch graphs of rational functions by identifying asymptotes, intercepts, and end behaviour.

Compose and invert functions, stating domain restrictions explicitly.

Differentiate from first principles and connect the limit definition to standard rules.

Apply product, quotient, and chain rules to composite and nested expressions.

Differentiate implicit relations and logarithmic and exponential forms.

Relate differentiability to continuity and identify points where a derivative fails to exist.

Integrate standard forms and recognise when substitution is required.

Evaluate definite integrals and apply the fundamental theorem of calculus correctly.

Use integration by parts and partial fractions on unfamiliar integrands.

Compute areas between curves and volumes of revolution, setting up limits carefully.

Solve optimisation and related-rates problems by translating word problems into derivative equations.

Find stationary points and classify them using first and second derivative tests.

Sketch curves using calculus features: monotonicity, concavity, and inflection points.

Combine a derivative and an integral in a single modelling problem.

Derive and apply the equation of a line and circle, including tangency and intersection conditions.

Use parametric and implicit descriptions of curves and convert between forms.

Solve trigonometric equations over a stated interval and express general solutions.

Apply compound-angle, double-angle, and sum-to-product identities to simplify expressions.

Use the sine and cosine rules to solve non-right triangles in context.

Identify arithmetic, geometric, and recursive sequences and find closed forms.

Test series for convergence using ratio and comparison reasoning, and sum geometric series.

Carry out matrix multiplication, inversion, and determinant computation on 2x2 and 3x3 cases.

Resolve vector problems in two and three dimensions, including dot and cross products.

Interpret the determinant as an area or volume scale factor.

Convert complex numbers between rectangular, polar, and exponential forms and plot them.

Apply De Moivre's theorem to powers and roots and locate roots geometrically.

Compute probabilities using conditional probability, independence, and Bayes-style reasoning.

Model discrete random variables and compute expectation and variance.

Interpret a normal distribution problem using standardisation and tables.

Construct direct proofs using deduction and counterexample, and distinguish the two.

Write a proof by contradiction and a proof by induction, each with a correct base case and inductive step.

Evaluate the validity of given arguments and identify logical fallacies.

Assemble a revision checklist covering every domain weighted by exam percentage.

Complete two full-length timed papers and review every error.