Indices & Surds
Index laws
aᵐ × aⁿ = aᵐ⁺ⁿ ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ ; (aᵐ)ⁿ = aᵐⁿ ; a⁻ⁿ = 1/aⁿ ; a^(1/n) = ⁿ√a Surds
√(ab) = √a · √b ; rationalise: 1/(a+√b) × (a−√b)/(a−√b) OCR A Level Mathematics A H240
🧮 OCR A Level Maths Formula Sheet 2026
Every formula you need for OCR A Level Mathematics A (H240) — pure mathematics, mechanics, and statistics — distilled into one organised reference for 2026 exams.
Pure Mathematics Mechanics Statistics H240 Specification
Our formula sheets are free to download — save this one as PDF for offline revision.
Aligned with the latest 2026 syllabus and board specifications. This sheet is prepared to match your exam board’s official specifications for the 2026 exam series.
Complete OCR A Level Maths Formulas in One Sheet
OCR A Level Mathematics A (H240) examines pure mathematics, mechanics, and statistics across three papers. This 2026 formula sheet groups every essential identity, rule, and equation by topic so you can revise efficiently and recall them under exam pressure.
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Pure: algebra, calculus, trig, vectors, numerical methods
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Mechanics: SUVAT, forces, momentum, moments, projectiles
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Statistics: distributions, correlation, hypothesis testing
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Aligned to OCR H240 specification for 2026 exams
Pure: Algebra, Polynomials & Sequences
Foundational manipulation skills you'll use across every paper.
Quadratics & Polynomials
Quadratic formula
x = [−b ± √(b² − 4ac)] / 2a Discriminant
Δ = b² − 4ac (Δ > 0 two real roots, Δ = 0 repeated, Δ < 0 none) Factor theorem
If f(a) = 0 then (x − a) is a factor of f(x) Remainder theorem
f(x) ÷ (x − a) leaves remainder f(a) Sequences & Series
Arithmetic
uₙ = a + (n−1)d ; Sₙ = n/2 [2a + (n−1)d] = n/2 (a + l) Geometric
uₙ = arⁿ⁻¹ ; Sₙ = a(1 − rⁿ)/(1 − r) ; S∞ = a/(1 − r) for |r| < 1 Binomial Expansion
Positive integer n
(a + b)ⁿ = Σ ⁿCᵣ aⁿ⁻ʳ bʳ where ⁿCᵣ = n!/(r!(n−r)!) General index
(1 + x)ⁿ = 1 + nx + n(n−1)/2! x² + n(n−1)(n−2)/3! x³ + … , valid for |x| < 1 Exponentials & Logarithms
y = aˣ ⇔ logₐ y = x ; ln x = logₑ x ; eˣ and ln x are inverse Log laws
log(xy) = log x + log y ; log(x/y) = log x − log y ; log xⁿ = n log x ; logₐ b = ln b / ln a Pure: Trigonometry
Identities, double-angle and the R-formula for harmonic combinations.
Pythagorean Identities
sin²θ + cos²θ = 1 ; 1 + tan²θ = sec²θ ; 1 + cot²θ = cosec²θ Compound & Double Angle
sin(A ± B) = sinA cosB ± cosA sinB cos(A ± B) = cosA cosB ∓ sinA sinB tan(A ± B) = (tanA ± tanB)/(1 ∓ tanA tanB) sin 2A = 2 sinA cosA ; cos 2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A ; tan 2A = 2 tanA/(1 − tan²A) R-formula (Harmonic Form)
a sinθ ± b cosθ = R sin(θ ± α) where R = √(a² + b²) and tan α = b/a (R > 0, 0 < α < π/2) Useful for solving equations and finding max/min of a sinθ + b cosθ.
Radian Measure & Small Angles
Arc s = rθ ; Sector area A = ½r²θ For small θ (radians): sinθ ≈ θ ; cosθ ≈ 1 − θ²/2 ; tanθ ≈ θ Pure: Differentiation
Standard derivatives, chain/product/quotient and implicit differentiation.
Standard Derivatives
d/dx (xⁿ) = n xⁿ⁻¹ ; d/dx (eˣ) = eˣ ; d/dx (ln x) = 1/x d/dx (sin x) = cos x ; d/dx (cos x) = −sin x ; d/dx (tan x) = sec²x d/dx (sec x) = sec x tan x ; d/dx (cosec x) = −cosec x cot x ; d/dx (cot x) = −cosec²x Chain, Product & Quotient Rules
Chain
dy/dx = (dy/du)(du/dx) Product
d/dx (uv) = u'v + uv' Quotient
d/dx (u/v) = (u'v − uv')/v² Implicit & Parametric
Implicit
Differentiate each term wrt x; for f(y), use df/dy · dy/dx Parametric
If x = f(t), y = g(t) then dy/dx = (dy/dt)/(dx/dt) Stationary Points
Set dy/dx = 0; classify with d²y/dx² (>0 min, <0 max) or sign change of dy/dx Pure: Integration
Standard integrals, substitution, parts, partial fractions and applications.
Standard Integrals
∫ xⁿ dx = xⁿ⁺¹/(n+1) + c (n ≠ −1) ; ∫ 1/x dx = ln|x| + c ∫ eˣ dx = eˣ + c ; ∫ sin x dx = −cos x + c ; ∫ cos x dx = sin x + c ∫ sec²x dx = tan x + c ; ∫ sec x tan x dx = sec x + c Integration by Substitution
∫ f(g(x)) g'(x) dx = ∫ f(u) du where u = g(x) Update limits when changing variable in a definite integral.
Integration by Parts
∫ u (dv/dx) dx = uv − ∫ v (du/dx) dx Choose u using LIATE: Logs, Inverse trig, Algebraic, Trig, Exponential.
Partial Fractions
1/((x−a)(x−b)) = A/(x−a) + B/(x−b) ; for repeated roots use A/(x−a) + B/(x−a)² Definite Integrals & Applications
Area under curve
A = ∫ₐᵇ y dx Volume of revolution
V = π ∫ₐᵇ y² dx (about x-axis) ; V = π ∫ₐᵇ x² dy (about y-axis) Pure: Vectors & Numerical Methods
3D vectors and numerical root-finding for non-elementary equations.
Vectors in 3D
a = (a₁, a₂, a₃) ; |a| = √(a₁² + a₂² + a₃²) Scalar product
a · b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b| cos θ Line equation
r = a + t b (vector form through point a, direction b) Newton–Raphson
xₙ₊₁ = xₙ − f(xₙ)/f'(xₙ) Fails if f'(xₙ) ≈ 0 or starting point is far from root.
Iteration xₙ₊₁ = g(xₙ)
Rearrange f(x) = 0 to x = g(x); iteration converges to root α if |g'(α)| < 1 Use a cobweb/staircase diagram to visualise convergence.
Trapezium Rule
∫ₐᵇ y dx ≈ h/2 [y₀ + yₙ + 2(y₁ + y₂ + … + yₙ₋₁)] where h = (b − a)/n Mechanics
Kinematics, dynamics, momentum, projectiles, moments and energy.
Kinematics (SUVAT)
v = u + at s = ut + ½at² s = vt − ½at² ; s = ½(u + v)t v² = u² + 2as Newton's Laws & Friction
F = ma (resultant force) Friction
F ≤ μR ; limiting friction F = μR (μ = coefficient of friction, R = normal reaction) Weight
W = mg, g = 9.8 m s⁻² (or 9.81) Momentum & Impulse
p = mv ; Impulse I = Ft = Δp = mv − mu Conservation
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂ (no external impulse) Projectiles
Components
uₓ = u cos α (constant) ; uᵧ = u sin α Range (level ground)
R = u² sin 2α / g ; Max height H = u² sin²α / (2g) Time of flight
T = 2u sin α / g Moments & Equilibrium
Moment
M = F × d (perpendicular distance) Equilibrium
ΣF = 0 and ΣM = 0 about any point Work, Energy & Power
Work W = Fs cos θ ; KE = ½mv² ; GPE = mgh Work–energy theorem
Net work = ΔKE Power
P = W/t = Fv Connected Particles
Treat the system as a whole for acceleration; use F = ma on each particle separately to find tension/normal force. Statistics
Sampling, summary measures, distributions and hypothesis testing.
Sampling & Summary Statistics
Sampling methods
Random, systematic, stratified, opportunity, quota, cluster Mean
x̄ = Σx/n ; from frequency table x̄ = Σfx/Σf Variance
σ² = Σ(x − x̄)²/n = Σx²/n − x̄² ; SD σ = √variance Correlation & Regression
Pearson's r
r = Sₓᵧ / √(Sₓₓ Sᵧᵧ) where Sₓᵧ = Σxy − (Σx)(Σy)/n Regression line
y = a + bx where b = Sₓᵧ/Sₓₓ and a = ȳ − b x̄ Probability
P(A ∪ B) = P(A) + P(B) − P(A ∩ B) ; P(A | B) = P(A ∩ B)/P(B) Independence
P(A ∩ B) = P(A) P(B) Discrete Random Variables
E(X) = Σ x P(X = x) ; Var(X) = Σ x² P(X = x) − [E(X)]² Binomial Distribution X ~ B(n, p)
P(X = r) = ⁿCᵣ pʳ (1 − p)ⁿ⁻ʳ Mean & variance
E(X) = np ; Var(X) = np(1 − p) Normal Distribution X ~ N(μ, σ²)
Standardise
Z = (X − μ)/σ ; Z ~ N(0, 1) Use the Φ tables or calculator to find P(Z < z).
Hypothesis Testing
Binomial test
H₀: p = p₀ vs H₁: p ≠ p₀ (two-tailed) or p > p₀ / p < p₀; compare P(X ≥ x | H₀) with significance level α Test for mean (known σ)
Z = (x̄ − μ)/(σ/√n), reject H₀ if |Z| > critical value How to Use This Formula Sheet
Boost your Cambridge exam confidence with these proven study strategies from our tutoring experts.
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Master the Formula Booklet
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Show Every Step
OCR mark schemes reward method marks. Lay out work clearly: state the formula, substitute values, then evaluate. This protects marks even if the final answer is wrong.
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Practise All Three Branches
Many students under-prepare for mechanics or statistics. Allocate revision evenly across pure, mechanics and statistics — the boundaries are tight.
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Formula Sheet FAQ
Quick answers about this free PDF and how to use it for exam revision and active recall.
Is the OCR A Level Maths Formula Sheet 2026 free to download as a PDF?
Yes. This Tutopiya formula sheet is free to use and you can download it as a PDF from this page for offline revision. There is no payment or account required for the PDF download.
What Mathematics topics and equations does this formula sheet cover?
This page groups key Mathematics formulas in one place for revision. Complete OCR A Level Mathematics A (H240) formula sheet for 2026: pure (algebra, calculus, trig, vectors), mechanics (SUVAT, momentum, projectiles), and statistics (distributions, hypothesis testing). Always cross-check with your official syllabus and past papers for your exam session.
Can I use this instead of the official exam formula booklet in the exam?
No. In the exam you must follow only what your exam board allows in the hall—usually the official formula booklet or data sheet where provided. This page is a revision and teaching aid, not a replacement for board-issued materials.
Who is this formula sheet for (Upper Secondary)?
It is written for students preparing for assessments at Upper Secondary in Mathematics, including classroom revision, homework support, and independent study. Teachers and tutors can also share it as a quick reference.
How should I revise with this formula sheet?
Work through past paper questions, quote the correct formula before substituting values, and check units and notation every time. Pair this sheet with timed practice and mark schemes so you see how examiners expect working to be set out.
Where can I get more help with Mathematics revision?
Explore Tutopiya’s study tools, past paper finder, and revision checklists linked from our tools hub, or book a trial lesson with a subject specialist for personalised support alongside this formula reference.
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This formula sheet aligns with OCR A Level Mathematics A (H240) for the 2026 exam series.
OCR provides a formula booklet in the exam — check the latest version on the OCR website to confirm which formulas are given and which must be learnt.