ZIMSEC · A Level
Further Mathematics
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Form 5 · Lower 6 4 topics
- Roots of polynomial equations For ax² + bx + c = 0 with roots α, β: α + β = −b/a; αβ = c/a.
- Rational functions and curve sketching For y = f(x) as a quotient, rearrange into a quadratic in x and use the discriminant ≥ 0 to find values y cannot take.
- Series and proof by induction Σr = n(n + 1)/2; Σr² = n(n + 1)(2n + 1)/6; Σr³ = n²(n + 1)²/4.
- Matrices and linear transformations Addition, multiplication (not commutative), identity I, determinant and inverse.
Form 6 · Upper 6 4 topicsExam year
- Polar coordinates A point is given by (r, θ): x = r cos θ, y = r sin θ, r² = x² + y².
- Further calculus: hyperbolic functions, reduction formulae and applications sinh x = (eˣ − e⁻ˣ)/2; cosh x = (eˣ + e⁻ˣ)/2; tanh x = sinh x/cosh x.
- Complex numbers and de Moivre's theorem (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ, for any integer n (proved by induction for positive n).
- Second-order differential equations Auxiliary equation am² + bm + c = 0: