Pure Mathematics: Algebra, Geometry, Trigonometry and Calculus
Pure Mathematics in Thanawiya Amma tests algebraic manipulation, spatial reasoning, trigonometric identities, and calculus. Marks depend on a correct method and an exact form, not only on the final number.
Core rules
- Complex numbers: i² = -1; modulus |a + bi| = √(a² + b²); the conjugate of a + bi is a - bi; divide by multiplying numerator and denominator by the conjugate.
- Differentiation: power, product, quotient and chain rules. Stationary points occur where f′(x) = 0; use f″(x) to test their nature.
- Integration: reverse the power rule; include + C for an indefinite integral; use a definite integral for area.
- Trigonometry: use identities to simplify before solving; work in radians in calculus unless the question states degrees.
- Geometry: vectors, dot product, equations of lines and planes, and conic sections.
Worked example
Differentiate f(x) = x²e^x. Use the product rule with u = x² and v = e^x. Then u′ = 2x and v′ = e^x. So f′(x) = 2x·e^x + x²·e^x = e^x(x² + 2x). The reasoning is to differentiate each factor once and add the two products; e^x is not a constant.
Traps
- Missing the chain rule in composite functions.
- Dropping + C in indefinite integration.
- Using degree mode when a calculus result requires radians.
- Squaring both sides and accepting extraneous roots.
- Sign errors in vector components or in the quotient rule.