Kuwait University Mathematics Aptitude Test cheat sheet

Mathematics

• Quadratic formula: for ax² + bx + c = 0 with a ≠ 0, x = (−b ± √(b² − 4ac)) / (2a); the discriminant b² − 4ac gives two real roots if positive, one if zero, and no real roots if negative.

• Laws of exponents: a^m · a^n = a^(m+n), a^m / a^n = a^(m−n), (a^m)^n = a^(mn), a^0 = 1 (a ≠ 0), a^(−n) = 1/a^n, and a^(1/n) = ⁿ√a.

• Laws of logarithms: log(xy) = log x + log y, log(x/y) = log x − log y, log(x^k) = k log x, log_b b = 1, log_b 1 = 0; change of base log_b x = ln x / ln b.

• Arithmetic progression: nth term a + (n−1)d; sum of n terms Sₙ = (n/2)(2a + (n−1)d) = (n/2)(first + last). Geometric progression: nth term ar^(n−1); Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1.

• Trigonometric identities: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, sin 2θ = 2 sin θ cos θ, cos 2θ = cos²θ − sin²θ = 1 − 2 sin²θ; sine rule a/sin A = b/sin B = c/sin C; cosine rule a² = b² + c² − 2bc cos A.

• Coordinate geometry: distance √((x₂−x₁)² + (y₂−y₁)²); midpoint ((x₁+x₂)/2, (y₁+y₂)/2); slope m = (y₂−y₁)/(x₂−x₁); parallel lines have equal slopes, perpendicular slopes multiply to −1.

• Circle with centre (h, k) and radius r: (x − h)² + (y − k)² = r²; expanding gives x² + y² + Dx + Ey + F = 0 with centre (−D/2, −E/2) and radius √((D/2)² + (E/2)² − F).

• Derivative rules: d/dx(xⁿ) = nx^(n−1), d/dx(sin x) = cos x, d/dx(cos x) = −sin x, d/dx(eˣ) = eˣ, d/dx(ln x) = 1/x; product rule (uv)′ = u′v + uv′; quotient rule (u/v)′ = (u′v − uv′)/v².

• Counting and probability: nPr = n!/(n−r)!, nCr = n!/(r!(n−r)!), P(A ∪ B) = P(A) + P(B) − P(A ∩ B), P(A|B) = P(A ∩ B)/P(B); independent events satisfy P(A ∩ B) = P(A)P(B).