Mathematics - Calculus, Vectors and Coordinate Geometry
The Scientific-stream mathematics paper covers limits and continuity, differentiation and its applications, integration, sequences and series, complex numbers, vectors, coordinate geometry including conic sections, matrices, and basic probability and statistics. Marks are awarded for method as well as answer, so every step must be visible and every formula stated before substitution.
Core rules
- A limit giving 0/0 is resolved by factorising and cancelling, or by multiplying by the conjugate when surds appear.
- Differentiation: power rule, product rule, quotient rule, chain rule; derivatives of sin x, cos x, e^x and ln x are standard results to memorise.
- Stationary points: solve f'(x) = 0; classify with f''(x) - f'' < 0 gives a local maximum, f'' > 0 a local minimum, f'' = 0 requires a sign test.
- Integration: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C for n ≠ −1; a definite integral equals F(b) − F(a).
- Vectors: two vectors are perpendicular when their dot product is zero; magnitude from the sum of squared components.
- Complex numbers: i² = −1; division uses the conjugate of the denominator.
Worked example
Find and classify the stationary points of f(x) = x³ − 3x² + 2. f'(x) = 3x² − 6x = 3x(x − 2), so stationary points occur at x = 0 and x = 2. f''(x) = 6x − 6. At x = 0: f'' = −6 < 0 → local maximum, value f(0) = 2. At x = 2: f'' = 6 > 0 → local minimum, value f(2) = −2. Reasoning: the first derivative locates the points, the second derivative classifies them, and substituting back gives the coordinates the question usually asks for.
Traps that cost marks
- Omitting + C in an indefinite integral.
- Losing the minus sign when differentiating cos x, or using degrees - calculus trigonometric derivatives require radians.
- Cancelling across a sum: (x + 2)/2 is not x.
- Giving only the x-value of a stationary point when the question asks for the point itself.