O-Level Maths Formula Sheet

E-Math (4052): formulas given in the exam paper

E-Math: formulas you must memorise

• Quadratic formula: x = (−b ± √(b² − 4ac)) ⁄ 2a
• Pythagoras’ theorem: c² = a² + b²
• Trigonometric ratios: sin = opp⁄hyp, cos = adj⁄hyp, tan = opp⁄adj; sin(180° − x) = sin x, cos(180° − x) = −cos x
• Gradient = (y₂ − y₁)⁄(x₂ − x₁); distance = √((x₂ − x₁)² + (y₂ − y₁)²); midpoint = ((x₁ + x₂)⁄2, (y₁ + y₂)⁄2)
• Circle: circumference 2πr, area πr²; cylinder: curved surface 2πrh, volume πr²h; prism volume = base area × height; pyramid volume = ¹⁄₃ base area × height
• Similar figures: ratio of areas = k², ratio of volumes = k³ for a length ratio k
• Angle properties of circles: angle at centre = 2 × angle at circumference; angles in the same segment are equal; angle in a semicircle = 90°; opposite angles of a cyclic quadrilateral sum to 180°; tangent ⊥ radius; tangents from an external point are equal
• Sum of interior angles of an n-sided polygon = (n − 2) × 180°; exterior angles sum to 360°
• Laws of indices: a^m × aⁿ = a^m+n, (a^m)ⁿ = a^mn, a⁻ⁿ = 1⁄aⁿ, a^1⁄n = ⁿ√a
• Probability: P(A or B) = P(A) + P(B) for mutually exclusive events; P(A and B) = P(A) × P(B) for independent events; P(not A) = 1 − P(A)
• Speed = distance⁄time; density = mass⁄volume; percentage change = change⁄original × 100%
• Median from cumulative frequency at ½n; interquartile range = Q₃ − Q₁

A-Math (4049): formulas given in the exam paper

A-Math: formulas you must memorise

• Discriminant: b² − 4ac > 0 two real roots, = 0 equal roots, < 0 no real roots; sum of roots = −b⁄a, product = c⁄a
• Remainder theorem: remainder when f(x) is divided by (x − a) is f(a); factor theorem: (x − a) is a factor if f(a) = 0
• Logarithms: log_a(xy) = log_ax + log_ay; log_a(x⁄y) = log_ax − log_ay; log_axⁿ = n log_ax; change of base log_ax = log_bx⁄log_ba
• R-formula: a sin θ ± b cos θ = R sin(θ ± α), R = √(a² + b²), tan α = b⁄a
• Basic identities: sin²A + cos²A = 1; tan A = sin A⁄cos A; sec A = 1⁄cos A, cosec A = 1⁄sin A, cot A = 1⁄tan A
• Differentiation: d⁄dx(xⁿ) = nxⁿ⁻¹; d⁄dx(sin x) = cos x; d⁄dx(cos x) = −sin x; d⁄dx(tan x) = sec²x; d⁄dx(e^x) = e^x; d⁄dx(ln x) = 1⁄x; chain, product and quotient rules
• Integration: ∫xⁿ dx = xⁿ⁺¹⁄(n+1) + c (n ≠ −1); ∫1⁄x dx = ln|x| + c; ∫e^x dx = e^x + c; ∫sin x dx = −cos x + c; ∫cos x dx = sin x + c; ∫sec²x dx = tan x + c
• Kinematics: v = ds⁄dt, a = dv⁄dt; displacement = ∫v dt; distance travelled needs the turning points where v = 0
• Coordinate geometry: perpendicular gradients multiply to −1; circle (x − a)² + (y − b)² = r², or x² + y² + 2gx + 2fy + c = 0 with centre (−g, −f) and radius √(g² + f² − c); area of a polygon by the shoelace method

H2 Math (9758)

A-Level candidates receive the MF26 list of formulae and statistical tables in the exam. It covers the pure-maths and statistics results a JC student is expected to apply, not memorise — our H2 Math tuition class trains students to recognise when each MF26 result is the shortcut the question is testing.

How to use this sheet

1. Separate “given” from “memorise”. Students lose marks by re-deriving formulas that are printed in the paper and by forgetting ones that are not.
2. Practise with real papers. Every formula above appears in the free model solutions on this site, worked in the exact format the marking scheme rewards.
3. Time it. In the last month, do one past paper a week under exam timing (2 h 15 min per paper for E-Math and A-Math).