Mathematics (including Further Mathematics)
STPM Mathematics rewards precise method as much as the final answer. The core covers functions, sequences, matrices, complex numbers, vectors, differentiation, integration, probability and statistics; Further Mathematics extends this into proof, matrices, differential equations and more demanding pure mathematics. Show every step: method marks are awarded for a correct substitution or a stated identity even when arithmetic slips.
Core rules
- Calculus: differentiate from first principles where asked; use chain, product and quotient rules; for definite integrals change limits when you substitute.
- Vectors: distinguish scalar and vector quantities; use the dot product for angles and perpendicularity.
- Complex numbers: use modulus–argument form for multiplication and de Moivre's theorem; check the quadrant.
- Probability: use conditional probability and independence correctly; draw a tree diagram for sequential events.
- Statistics: know when to use binomial, Poisson and normal approximations, and state hypotheses before testing.
Worked example
Evaluate ∫ x e^x dx. Use integration by parts: let u = x and dv = e^x dx, so du = dx and v = e^x. Then ∫ x e^x dx = x e^x − ∫ e^x dx = x e^x − e^x + C. Differentiate the result to check: d/dx (x e^x − e^x) = (e^x + x e^x) − e^x = x e^x. The constant C is part of the answer.
Traps
Dropping '+ C'; sign errors in integration by parts; using degrees in calculus; forgetting to change limits after substitution; giving a probability greater than 1; rounding too early.