Mathematics (Natural Science Stream)
The mathematics paper for the natural science stream tests algebraic fluency, calculus, and vector/statistics reasoning from Grades 9–12. Most items are multi-step but short; marks are lost at transition points - a sign, a domain restriction, or an un-eliminated parameter.
Functions, limits, and continuity
- A function is defined only on its natural domain; check denominators, even roots, and log arguments before simplifying.
- lim_{x→a} f(x) exists only if left and right limits agree. Factor or rationalise to remove 0/0 forms.
Differentiation
- Power, product, quotient, chain. d/dx [f(g(x))] = f'(g(x))g'(x).
- Increasing where f' > 0; stationary points where f' = 0; classify with sign change or f''.
Integration
- Indefinite integrals need +C. Definite integrals use the fundamental theorem: ∫_a^b f(x) dx = F(b) − F(a).
- Area below the x-axis counts as negative unless the question asks for total area.
Vectors and sequences
- Dot product gives angle and perpendicularity (a·b = 0); cross product gives a perpendicular vector and area.
- Arithmetic: a_n = a + (n−1)d; geometric: a_n = ar^{n−1}; sum to infinity exists only when |r| < 1.
Worked example
Differentiate f(x) = x²e^{3x}. This is a product with an inner function. Let u = x², v = e^{3x}. Then u' = 2x and v' = 3e^{3x} by the chain rule. Product rule: f' = u'v + uv' = 2x e^{3x} + x²(3e^{3x}) = x e^{3x}(2 + 3x). Check: the common factor x e^{3x} is correct; the bracket is (2 + 3x), not (2 + 3x²).
Traps
- Dropping +C in an indefinite integral.
- Using degrees in a calculus or trigonometric limit question when radians are required.
- Cancelling a factor that can be zero, then losing a solution.
- In matrices, writing A⁻¹ without checking det A ≠ 0.
- Confusing the sum of an arithmetic series with the nth term.