TCE Mathematics Methods past exams 2021–2025, by year and topic

The rest of the syllabus

Question-level mapping for these areas is still being verified. Every official paper covering them is in the year table above.

• Function notation, domains and ranges — A function is a rule with an input set · Finding a maximal real domain · Range is an output set, not another domain · Reading domain and range from a graph · Domains in practical models · Communicating sets and checking conclusions
• Polynomial factorisation, binomial expansion and graph features — Factors connect algebra to intercepts · Factor and remainder reasoning · Multiplicity and sign charts · Using the binomial theorem efficiently · Building a graph from algebra · Parameters and constructing polynomial models
• Exponential and logarithmic functions and modelling — Exponential change and the role of the base · Logarithms reverse exponential rules · Solving equations with logarithms · Graphs, asymptotes and transformations · Estimating an exponential model from data · Decay toward a baseline and threshold times
• Transformations, composites, inverses and piecewise functions — Tracking points through transformations · Composing functions and checking existence · Finding an inverse with a restricted domain · Reciprocal functions and inherited holes · Piecewise definitions and boundary values · Combining domains, inverses and transformed rules
• Radians, the unit circle and exact values — Radians measure angle through arc length · The unit circle defines sine and cosine · Deriving the standard exact values · Reference angles and coterminal rotations · Using the Pythagorean identity with quadrant data · Exact evaluation, notation and meaningful checks
• Trigonometric identities and symmetry — An identity holds across its domain · Reflection and rotation identities · Quarter-turn relationships · Simplifying expressions without losing restrictions · Using identities to solve equations · Checking identities and detecting false rules
• Circular graphs, transformations and equations — Reading the parameters of sine and cosine · Tangent graphs and asymptotes · Solving basic circular equations on an interval · Transforming the domain before solving · Equations with both sine and cosine · Checking graphs and numerical roots
• Periodic modelling and interpretation — Choosing a sinusoidal model · Determining phase from an event · Recovering a period from incomplete cycle information · Solving a threshold and finding a duration · Comparing changed periodic conditions · Interpreting a model within its limits
• Limits, first principles and differentiability — Average and instantaneous rates · What a limit does and does not say · Deriving the quadratic derivative from first principles · Cubic first principles and interpreting the result · Continuity is necessary but not sufficient · Derivative graphs and local behaviour
• Product, quotient and chain rules — Elementary derivatives and linear combinations · The chain rule follows nested dependence · Products require two contributions · Quotients and the order of subtraction · Combining rules in a structured calculation · Verification and using a derivative result
• Tangents, normals and derivative graphs — A tangent needs a point and a gradient · Normals and exceptional gradients · Finding where a tangent has a specified direction · Tangents through an external point · Constructing a derivative graph from known slopes · Using a tangent as a local approximation
• Stationary points, optimisation and motion — Stationary candidates and sign changes · Local extrema versus interval extrema · Building an optimisation function from constraints · Optimising a volume with a meaningful domain · Position, velocity and acceleration · Distance, displacement and a complete motion account
• Antiderivatives and boundary conditions — Antidifferentiation recovers a family of functions · The power rule and its exception · Linear inner functions and reverse chain factors · Determining a constant from a point · Recovering unknown parameters with multiple conditions · Reconstructing position from velocity
• Definite integrals and the fundamental theorem — Accumulation and signed contributions · Evaluating with an antiderivative · Additivity, reversed limits and constant factors · Accumulation functions and changing upper limits · Solving for an unknown boundary or parameter · Using a supplied derivative relationship
• Areas under and between curves — Geometric area and signed integral · Area below one positive curve · Area between two curves uses their vertical difference · Splitting where the upper curve changes · Areas with a parameter · Area models, units and final verification
• Recovering functions and modelling displacement — A rate determines change before it determines level · Recovering a curve from tangent information · Displacement from a velocity function · Reading displacement from a velocity graph · Piecewise rates and continuity of accumulated quantity · Average rate and checking a reconstructed model
• Discrete random variables, expectation and variance — A random variable assigns numbers to outcomes · Completing and using a probability table · Expected value is a weighted long-run mean · Variance measures spread around the mean · Comparing and transforming scores · Interpreting distributions and avoiding overclaims
• Binomial distributions and modelling assumptions — Recognising a binomial experiment · Exact counts and the binomial formula · Cumulative events and complements · Mean, variance and the shape of a count · Solving for trial numbers or probabilities · Checking independence and interpreting expected counts
• Normal probabilities, quantiles and inverse parameters — A continuous normal model · Standardising and interpreting z-scores · Calculating interval and tail probabilities · Quantiles and inverse normal calculations · Finding an unknown mean or standard deviation · Continuous density and model checks
• Sample proportions and confidence intervals — Population parameters and sample statistics · The sampling distribution of a proportion · Probabilities for sample proportions · Constructing a confidence interval · Interpreting confidence correctly · Confidence, precision and sample size