Algebra
Quadratic formula
x = (−b ± √(b² − 4ac)) / 2a For ax² + bx + c = 0
Difference of two squares
a² − b² = (a − b)(a + b) Sum of an arithmetic progression
Sₙ = n/2 [2a + (n − 1)d] a = first term, d = common difference
nth term of an AP
Tₙ = a + (n − 1)d nth term of a GP
Tₙ = arⁿ⁻¹ r = common ratio
Sum of a GP
Sₙ = a(rⁿ − 1)/(r − 1) Sum to infinity of a GP
S∞ = a/(1 − r) valid when |r| < 1
Laws of indices
aᵐ × aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ Logarithm laws
logₐ(MN) = logₐM + logₐN logₐ(M/N) = logₐM − logₐN
Change of base
logₐb = log_c b / log_c a Binomial expansion
(a + b)ⁿ = Σ ₙCᵣ aⁿ⁻ʳ bʳ Geometry and Trigonometry
Area of a circle
A = πr² Circumference of a circle
C = 2πr Area of a triangle
A = ½ × base × height Area of a triangle (Heron)
A = √[s(s−a)(s−b)(s−c)] s = semi-perimeter
Area of a sector
A = (θ/360°) × πr² Length of an arc
l = (θ/360°) × 2πr Pythagoras’ theorem
a² + b² = c² Volume of a cylinder
V = πr²h Volume of a sphere
V = 4/3 πr³ Volume of a cone
V = ⅓ πr²h Sine rule
a/sin A = b/sin B = c/sin C Cosine rule
a² = b² + c² − 2bc cos A Area of a triangle (trig)
A = ½ ab sin C Basic ratios
sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj Identity
sin²θ + cos²θ = 1 Calculus
Power rule (differentiation)
d/dx (xⁿ) = nxⁿ⁻¹ Derivative of sin/cos
d/dx(sin x) = cos x, d/dx(cos x) = −sin x Product rule
d/dx(uv) = u′v + uv′ Quotient rule
d/dx(u/v) = (u′v − uv′)/v² Power rule (integration)
∫ xⁿ dx = xⁿ⁺¹/(n+1) + C Integral of sin/cos
∫ sin x dx = −cos x + C, ∫ cos x dx = sin x + C Statistics
Mean
x̄ = Σx / n Median (grouped)
L + [(n/2 − F)/f] × c Standard deviation
σ = √(Σ(x − x̄)² / n) Variance
σ² = Σ(x − x̄)² / n Permutation
ⁿPᵣ = n!/(n − r)! Combination
ⁿCᵣ = n!/[r!(n − r)!] Probability
P(A) = n(A)/n(S) Probability (independent)
P(A∩B) = P(A) × P(B) Probability (mutually exclusive)
P(A∪B) = P(A) + P(B) Test yourself on Mathematics
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