Mathematics - compulsory written examination
What the mathematics examination tests
Mathematics is a compulsory written examination. It tests the ability to select and apply methods across the syllabus: functions and graphs, limits and derivatives, integration, sequences, vectors and geometry, and probability and statistics, with the exact balance depending on the course variant. Marks are awarded for justified method as well as final answers, so a correct result without working can score poorly.
Core rules and concepts
- Show the method: state the formula, substitute, simplify, and give the result with units or domain restrictions where relevant.
- For a function study, the standard sequence is domain, parity and periodicity where relevant, limits and asymptotes, derivative and sign table, extrema, second derivative and concavity, and a sketch.
- In integration, check whether a substitution or integration by parts is signalled by the integrand; do not forget the constant or the evaluation limits.
- In probability, define the event and the sample space before calculating; use the complement when it shortens the work.
- Justify with a sentence: "since f'(x) < 0 on (0, 2), f is decreasing there".
Worked example
Find the stationary points of f(x) = x³ − 3x and classify them.
Reasoning: f'(x) = 3x² − 3 = 3(x − 1)(x + 1), so f'(x) = 0 at x = 1 and x = −1. Sign table: f' > 0 for x < −1, f' < 0 for −1 < x < 1, f' > 0 for x > 1. Hence there is a local maximum at x = −1, f(−1) = 2, and a local minimum at x = 1, f(1) = −2. A bare answer "maximum at −1, minimum at 1" loses the justification marks; the sign table is the evidence.
Traps
- Dropping the sign or the constant during integration.
- Forgetting domain restrictions (for example, ln x requires x > 0, and denominators cannot be zero).
- Using a calculator result without explaining the method.
- Rounding too early and carrying an error into the final answer.