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O-Level A-Math Formula Sheet

Advanced formulas for Additional Mathematics mastery.

Algebra & Logarithms

  • • Discriminant (b²-4ac): >0 (2 roots), =0 (equal), <0 (no real roots)
  • • Sum of Roots (α+β) = -b/a | Product (αβ) = c/a
  • • Log Laws: log(AB) = logA + logB | log(A/B) = logA - logB | log(Aⁿ) = nlogA
  • • Change of Base: logₐb = (log꜀b) / (log꜀a)
  • • Indices: aᵐ × aⁿ = aᵐ⁺ⁿ | (aᵐ)ⁿ = aᵐⁿ | a⁻ⁿ = 1/aⁿ

Trigonometry Identities

  • • sin²A + cos²A = 1 | 1 + tan²A = sec²A | 1 + cot²A = cosec²A
  • • Double Angle: sin2A = 2sinAcosA | cos2A = cos²A - sin²A = 2cos²A - 1
  • • Addition Formula: sin(A±B) = sinAcosB ± cosAsinB
  • • R-Formula: a sinθ ± b cosθ = R sin(θ ± α)

Calculus (Differentiation)

  • • Power Rule: d/dx(axⁿ) = anxⁿ⁻¹
  • • Product Rule: u v' + v u' | Quotient Rule: (v u' - u v') / v²
  • • Chain Rule: dy/dx = (dy/du) × (du/dx)
  • • Trig: d/dx(sin x) = cos x | d/dx(cos x) = -sin x | d/dx(tan x) = sec²x
  • • Stationary Points: dy/dx = 0 (Max if d²y/dx² < 0, Min if d²y/dx² > 0)

Calculus (Integration)

  • • ∫xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ -1)
  • • ∫ (1/x) dx = ln|x| + C
  • • ∫ eᵏˣ dx = (1/k)eᵏˣ + C
  • • ∫ sin kx dx = -(1/k)cos kx + C | ∫ cos kx dx = (1/k)sin kx + C
  • • Area under curve = ∫[a to b] y dx

Coordinate Geometry & Binomial

  • • Distance Formula: √[(x₂-x₁)² + (y₂-y₁)²]
  • • Perpendicular Lines: m₁ × m₂ = -1
  • • Binomial Theorem: (a+b)ⁿ = aⁿ + ⁿC₁aⁿ⁻¹b + ... + ⁿCᵣaⁿ⁻ʳbʳ + ... + bⁿ
  • • Linear Law: Transform non-linear to Y = mX + c
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