In the formula booklet (MF19): pure mathematics
Our own selection of the main formulae in MF19, marked with the paper they belong to. Everything unmarked is used from AS (Papers 1, 2, 4 and 5).
| Quantity | Formula |
|---|---|
| Arc length (θ in radians) | s = rθ |
| Sector area (θ in radians) | A = ½r²θ |
| Quadratic formula | x = (−b ± √(b² − 4ac)) ÷ 2a |
| Arithmetic series | un = a + (n − 1)d, Sn = ½n{2a + (n − 1)d} |
| Geometric series | un = arⁿ⁻¹, Sn = a(1 − rⁿ) ÷ (1 − r) |
| Sum to infinity, |r| < 1 | S∞ = a ÷ (1 − r) |
| Binomial coefficient | ⁿCr = n! ÷ (r!(n − r)!) |
| Binomial series, rational n, |x| < 1 Paper 3 | (1 + x)ⁿ = 1 + nx + n(n − 1)x²/2! + … |
| Pythagorean identities | sin²θ + cos²θ ≡ 1, 1 + tan²θ ≡ sec²θ |
| Compound angles Paper 3 | sin(A ± B) ≡ sinA cosB ± cosA sinB |
| Double angles Paper 3 | sin2A ≡ 2sinA cosA, cos2A ≡ 1 − 2sin²A |
| Product rule Paper 3 | d(uv)/dx = v du/dx + u dv/dx |
| Quotient rule Paper 3 | d(u/v)/dx = (v du/dx − u dv/dx) ÷ v² |
| Parametric differentiation Paper 3 | dy/dx = (dy/dt) ÷ (dx/dt) |
| Integration by parts Paper 3 | ∫u dv/dx dx = uv − ∫v du/dx dx |
| Logarithmic integral Paper 3 | ∫f′(x)/f(x) dx = ln|f(x)| + c |
| Scalar product Paper 3 | a·b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b|cosθ |
In the formula booklet: mechanics
| Quantity | Formula |
|---|---|
| Constant acceleration | v = u + at, s = ut + ½at² |
| Constant acceleration | v² = u² + 2as, s = ½(u + v)t |
In the formula booklet: probability & statistics
| Quantity | Formula |
|---|---|
| Mean and standard deviation | x̄ = Σx/n, σ = √(Σx²/n − x̄²) |
| Discrete random variable | E(X) = Σxp, Var(X) = Σx²p − {E(X)}² |
| Binomial B(n, p) | P(X = r) = ⁿCr pʳ(1 − p)ⁿ⁻ʳ, mean np, variance np(1 − p) |
| Geometric Geo(p) | P(X = r) = p(1 − p)ʳ⁻¹, mean 1/p |
| Poisson Po(λ) Paper 6 | P(X = r) = e−λλʳ ÷ r!, mean λ, variance λ |
| Unbiased variance estimate Paper 6 | s² = (Σx² − (Σx)²/n) ÷ (n − 1) |
| Sample mean (Central Limit Theorem) Paper 6 | X̄ ~ N(μ, σ²/n) |
Not in the booklet: learn these
| Quantity | Formula |
|---|---|
| Friction | F ≤ μR (F = μR when sliding) |
| Weight | W = mg, with g = 10 m s⁻² |
| Newton's second law | F = ma |
| Work done | W = Fd cosθ |
| Kinetic and potential energy | KE = ½mv², PE = mgh |
| Power | P = Fv |
| Momentum | p = mv (conserved in collisions) |
| Standardising | Z = (X − μ) ÷ σ |
| Normal approximation to binomial | use when np > 5 and nq > 5 |
| Trapezium rule Paper 3 | ½h{y₀ + 2(y₁ + … + yn−1) + yn} |
| Linear transformation Paper 6 | E(aX + b) = aE(X) + b, Var(aX + b) = a²Var(X) |
The papers
| Paper | Content | Length, marks |
|---|---|---|
| Paper 1 | Pure Mathematics 1 | 1 h 50 min, 75 |
| Paper 2 | Pure Mathematics 2 (AS only) | 1 h 15 min, 50 |
| Paper 3 | Pure Mathematics 3 | 1 h 50 min, 75 |
| Paper 4 | Mechanics | 1 h 15 min, 50 |
| Paper 5 | Probability & Statistics 1 | 1 h 15 min, 50 |
| Paper 6 | Probability & Statistics 2 | 1 h 15 min, 50 |
A Level routes: Papers 1, 3, 4 and 5, or Papers 1, 3, 5 and 6. Graphical and symbolic calculators are not allowed; give answers to 3 significant figures (angles to 1 decimal place).
How to use this sheet
- Practise with MF19 in front of you so you know where each formula is.
- Learn the formulae in the last table: they are not in the booklet.
- Use g = 10 m s⁻² in mechanics unless the question says otherwise.
- Source: the official Cambridge syllabus (PDF). Cambridge's own booklet is copyright, so this is our own layout of the standard formulae.