Cambridge · O Level · Syllabus 4037
Additional Mathematics
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Form 3 5 topics
- Functions Domain, range, one-one and many-one mappings; composite functions fg(x) = f(g(x)); inverse functions f⁻¹ (only for one-one functions); the graphs of f and f⁻¹ are reflections in y = x.
- Quadratics, inequalities and polynomials Completing the square: ax² + bx + c = a(x + p)² + q gives the vertex (−p, q). Discriminant b² − 4ac decides the number of real roots; conditions for a line to meet, be a tangent to, or miss a curve.
- Surds, indices, logarithms and exponentials Simplify √50 = 5√2; rationalise denominators: 1/(√3 − 1) = (√3 + 1)/2.
- Coordinate geometry and circles Gradient, midpoint and distance formulae; equations y − y₁ = m(x − x₁); parallel (equal gradients) and perpendicular lines (m₁m₂ = −1); perpendicular bisectors; area of a triangle or polygon using coordinates (shoelace method).
- Circular measure and trigonometry π rad = 180°; arc length s = rθ; sector area A = ½r²θ.
Form 4 4 topicsExam year
- Permutations, combinations and series ⁿPᵣ = n!/(n − r)! (order matters); ⁿCᵣ = n!/(r!(n − r)!) (order does not matter); arrangements with restrictions (together, apart, ends).
- Vectors in two dimensions Column vectors and i, j notation; magnitude |xi + yj| = √(x² + y²); unit vectors; position vectors; parallel vectors are multiples of each other.
- Differentiation and its applications d/dx(xⁿ) = nxⁿ⁻¹; d/dx(eᵃˣ) = aeᵃˣ; d/dx(ln x) = 1/x; d/dx(sin x) = cos x; d/dx(cos x) = −sin x; d/dx(tan x) = sec²x. Chain, product and quotient rules.
- Integration and kinematics Reverse of differentiation: ∫xⁿ dx = xⁿ⁺¹/(n + 1) + c (n ≠ −1); ∫eᵃˣ dx = eᵃˣ/a + c; ∫1/x dx = ln x + c; ∫sin ax dx = −cos ax/a + c; ∫cos ax dx = sin ax/a + c; ∫(ax + b)ⁿ dx = (ax + b)ⁿ⁺¹/[a(n + 1)] + c.