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A-Math 'Formula Bible' (Syllabus 4049)

The advanced reference for A-Math excellence. Every identity, logarithm law, circle property, and calculus rule you need for Paper 1 & 2.

The A-Math Formula Bible is an advanced reference for Secondary 3 and 4 students taking GCE O-Level Additional Mathematics (Syllabus 4049). It covers every identity, rule, and formula across Algebra & Logarithms, Trigonometric Identities, Calculus (Differentiation and Integration), and Coordinate Geometry & Binomial Theorem. Designed to match the cognitive flow of actual O-Level questions.

Algebra & Logarithms

Quadratic Discriminant (b² - 4ac): > 0 (2 distinct real roots / cuts x-axis twice), = 0 (2 equal real roots / tangent to x-axis), < 0 (no real roots / does not cut x-axis). 'Always positive/negative' quadratic: a > 0 & b² - 4ac < 0 (above x-axis) or a < 0 & b² - 4ac < 0 (below x-axis)

Roots of Quadratic Equations: α + β = -b/a and αβ = c/a. New quadratic equation with roots p and q: x² - (p+q)x + pq = 0

Logarithm Rules: log_a(xy) = log_a x + log_a y | log_a(x/y) = log_a x - log_a y | log_a(xⁿ) = n log_a x. Common trap: log(x+y) != log x + log y and log(x)/y != log(x) - log(y)

Logarithm Change of Base: log_a b = log_c b / log_c a | Special case: log_a b = 1 / log_b a. Note: ln x = log_e x and e^(ln x) = x

Indices Rules: aᵐ × aⁿ = aᵐ⁺ⁿ | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | (aᵐ)ⁿ = aᵐⁿ | a⁻ⁿ = 1/aⁿ | a^(m/n) = ⁿ√aᵐ. Remember: a⁰ = 1 (for a != 0)

Trigonometry Identities

Principal Identities: sin² A + cos² A = 1 | 1 + tan² A = sec² A | 1 + cot² A = csc² A

Double Angle Formulas (DAF): sin 2A = 2 sin A cos A | cos 2A = cos² A - sin² A = 2 cos² A - 1 = 1 - 2 sin² A | tan 2A = 2 tan A / (1 - tan² A)

Addition Formulas: sin(A ± B) = sin A cos B ± cos A sin B | cos(A ± B) = cos A cos B ∓ sin A sin B | tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B)

R-Formula: a sin θ ± b cos θ = R sin(θ ± α) and a cos θ ± b sin θ = R cos(θ ∓ α) where R = √(a² + b²) and tan α = b/a (where a, b > 0)

Calculus (Differentiation)

Chain Rule, Product Rule & Quotient Rule: Chain Rule: dy/dx = dy/du × du/dx. Product Rule: d/dx(uv) = u(dv/dx) + v(du/dx). Quotient Rule: d/dx(u/v) = (v(du/dx) - u(dv/dx)) / v²

Derivatives of Trigonometric Functions: d/dx(sin f(x)) = f'(x) cos f(x) | d/dx(cos f(x)) = -f'(x) sin f(x) | d/dx(tan f(x)) = f'(x) sec² f(x). Note: Angles must be in radians!

Exponential & Logarithmic Derivatives: d/dx(e^(f(x))) = f'(x) e^(f(x)) | d/dx(ln f(x)) = f'(x) / f(x)

Stationary Points & Nature: Find dy/dx and solve for dy/dx = 0. Determine nature using second derivative test: d²y/dx² > 0 (Minimum point), d²y/dx² < 0 (Maximum point). If d²y/dx² = 0, use the first derivative sign table test (inflection point)

Calculus (Integration)

Integration of Algebraic Functions: ∫ (ax+b)ⁿ dx = (ax+b)ⁿ⁺¹ / (a(n+1)) + C (for n != -1). For n = -1: ∫ 1/(ax+b) dx = 1/a ln|ax+b| + C

Integration of Exponential & Trigonometric Functions: ∫ e^(ax+b) dx = 1/a e^(ax+b) + C | ∫ sin(ax+b) dx = -1/a cos(ax+b) + C | ∫ cos(ax+b) dx = 1/a sin(ax+b) + C | ∫ sec²(ax+b) dx = 1/a tan(ax+b) + C

Definite Integrals & Area Under Curve: ∫[a to b] f(x) dx = F(b) - F(a). Area between curve and x-axis = ∫[a to b] |y| dx. Area between curve and y-axis = ∫[c to d] |x| dy

Kinematics: Displacement s, Velocity v = ds/dt, Acceleration a = dv/dt = d²s/dt². Reversing: v = ∫ a dt and s = ∫ v dt. 'Instantaneous rest' means v = 0. 'Initial state' means t = 0

Coordinate Geometry & Binomial

Distance, Midpoint & Gradients: Distance = √[(x₂-x₁)² + (y₂-y₁)²] | Midpoint = ((x₁+x₂)/2, (y₁+y₂)/2). Parallel lines: m₁ = m₂. Perpendicular lines: m₁ × m₂ = -1

Circle Equations: Standard form: (x-h)² + (y-k)² = r² with center (h,k) and radius r. General form: x² + y² + 2gx + 2fy + c = 0 with center (-g, -f) and radius √(g²+f²-c)

Linear Law: Transform non-linear relation to Y = mX + c form. E.g., y = axᵇ => ln y = ln a + b ln x (plot ln y vs ln x, gradient m = b, Y-intercept c = ln a)

Binomial Theorem: (a+b)ⁿ = aⁿ + ⁿC₁ aⁿ⁻¹ b + ⁿC₂ aⁿ⁻² b² + ... + bⁿ. General term T_r+1 = ⁿC_r a^(n-r) b^r. Note: ⁿC_r = n! / (r!(n-r)!)

Frequently Asked Questions

What A-Math topics does this formula sheet cover?▾

It covers 45 formulas across 5 domains: Algebra & Logarithms (Discriminant, Sum/Product of Roots, Log Laws, Change of Base, Indices), Trigonometry Identities (Pythagorean, Double Angle, Addition Formula, R-Formula), Calculus Differentiation (Power, Product, Quotient, Chain rules), Calculus Integration (Polynomial, Exponential, Trigonometric), and Coordinate Geometry & Binomial Theorem.

How is this different from the E-Math checklist?▾

The E-Math checklist covers Elementary Mathematics (Syllabus 4048) and is a self-assessment tool. This A-Math Formula Bible covers Additional Mathematics (Syllabus 4049) — a separate, more advanced syllabus — and is designed as a rapid-recall reference sheet for exam preparation.

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