HKDSE Maths (Core) cheat sheet

Quadratic formula: for ax² + bx + c = 0 (a ≠ 0), x = [−b ± √(b² − 4ac)] / (2a); the discriminant Δ = b² − 4ac gives two distinct real roots (Δ > 0), one repeated real root (Δ = 0), or no real roots (Δ < 0).

Sum and product of roots: if α, β are roots of ax² + bx + c = 0, then α + β = −b/a and αβ = c/a; conversely x² − (sum)x + (product) = 0.

Laws of indices: a^m · a^n = a^(m+n), a^m ÷ a^n = a^(m−n), (a^m)^n = a^(mn), (ab)^n = a^n b^n, a^0 = 1 (a ≠ 0), a^(−n) = 1/a^n, a^(1/n) = ⁿ√a.

Laws of logarithms (base a > 0, a ≠ 1; arguments positive): log(MN) = log M + log N, log(M/N) = log M − log N, log M^n = n log M, log_a a = 1, log_a 1 = 0, and change of base log_a M = log_b M ÷ log_b a.

Remainder and factor theorems: dividing polynomial P(x) by (x − a) leaves remainder P(a); (x − a) is a factor of P(x) if and only if P(a) = 0.

Percentage change = (new − old)/old × 100%; successive changes multiply, e.g. +20% then −20% gives ×1.2×0.8 = 0.96, a net 4% decrease. Simple interest I = PRT/100; compound amount A = P(1 + r%)^n.

Arithmetic sequence: T(n) = a + (n − 1)d, sum S(n) = n/2 [2a + (n − 1)d] = n/2 (a + l). Geometric sequence: T(n) = ar^(n−1), sum S(n) = a(1 − r^n)/(1 − r) for r ≠ 1, and S(∞) = a/(1 − r) only when |r| < 1.

Variation: direct y ∝ x means y = kx; inverse y ∝ 1/x means y = k/x; joint/partial variation combines these, and k is found from one data pair then applied to the required case.

Inequalities: multiplying or dividing both sides by a negative number reverses the inequality sign; for a quadratic inequality, find the roots first, then test intervals or sketch the parabola to read off the solution set.

Angles: angles on a straight line sum to 180°; angles at a point sum to 360°; vertically opposite angles are equal; for parallel lines, corresponding and alternate angles are equal and co-interior angles sum to 180°.

Triangle and polygon facts: exterior angle of a triangle equals the sum of the two interior opposite angles; interior angle sum of an n-sided polygon = (n − 2) × 180°, and exterior angle sum = 360°.

Congruence conditions (SSS, SAS, ASA, AAS, RHS) and similarity conditions (AAA, three sides proportional, two sides proportional with included angle equal); in similar figures, areas are in the square of the ratio and volumes in the cube.

Circle theorems: angle in a semicircle is 90°; angle at the centre is twice the angle at the circumference on the same arc; angles in the same segment are equal; opposite angles of a cyclic quadrilateral sum to 180°; tangent ⊥ radius at the point of contact; tangents from an external point are equal.

Pythagoras' theorem a² + b² = c²; trigonometric ratios SOH-CAH-TOA; exact values sin 30° = 1/2, cos 60° = 1/2, tan 45° = 1, sin 45° = cos 45° = √2/2, sin 60° = cos 30° = √3/2.

Sine rule a/sin A = b/sin B = c/sin C; cosine rule c² = a² + b² − 2ab cos C; area of triangle = (1/2)ab sin C. Bearings are measured clockwise from north and written with three digits.

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 lines have slopes whose product is −1.

Equation of a circle with centre (h, k) and radius r: (x − h)² + (y − k)² = r²; the general form x² + y² + Dx + Ey + F = 0 has centre (−D/2, −E/2) and radius √[(D/2)² + (E/2)² − F].

Statistics: mean = sum ÷ count; median is the middle value of ordered data; mode is the most frequent value; range = largest − smallest; for grouped data use class mid-values. Adding a constant to every datum shifts the mean and median but leaves the standard deviation unchanged; multiplying every datum by k multiplies the standard deviation by |k|.

Probability: P(E) = favourable outcomes ÷ total equally likely outcomes; P(not E) = 1 − P(E); P(A or B) = P(A) + P(B) − P(A and B); for independent events P(A and B) = P(A) × P(B).