2×2 Matrix Determinant & Inverse
A = (a b; c d) ; det A = ad − bc A⁻¹ = (1/det A) (d −b; −c a) , exists iff det A ≠ 0 OCR A Level Further Mathematics A H245
🔢 OCR A Level Further Mathematics Formula Sheet 2026
All the core pure, further mechanics, and further statistics formulas for OCR A Level Further Maths A (H245) — collected in one organised reference for 2026 exams.
Core Pure Further Mechanics Further Statistics H245 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.
Master OCR Further Maths Across All Papers
OCR A Level Further Mathematics A (H245) builds on H240 with rigorous core pure content and options in further mechanics and statistics. This 2026 sheet condenses every key formula, identity and theorem into one navigable reference.
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Matrices, eigenvalues and transformations
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Complex numbers, De Moivre and roots of unity
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Polar curves, hyperbolic functions, further calculus
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Further mechanics: SHM, circular motion, Hooke's law
Matrices
Determinants, inverses, transformations and eigenvalues.
3×3 Determinant
Expand along row 1: det A = a(ei − fh) − b(di − fg) + c(dh − eg) Linear Transformations (2D)
Rotation by θ
R = (cosθ −sinθ; sinθ cosθ) Reflection in y = x tanθ
(cos 2θ sin 2θ; sin 2θ −cos 2θ) Enlargement (factor k)
(k 0; 0 k) Eigenvalues & Eigenvectors
Av = λv with v ≠ 0 ; characteristic equation det(A − λI) = 0 Diagonalisation: A = PDP⁻¹ where columns of P are eigenvectors and D = diag(λ₁, λ₂, …).
Invariant Lines & Points
Invariant point: A x = x ⇒ (A − I)x = 0 Invariant line y = mx + c maps onto itself; substitute and equate. Complex Numbers
Argand diagrams, modulus-argument, De Moivre and roots of unity.
Forms of a Complex Number
Cartesian
z = x + iy with i² = −1 Modulus & argument
|z| = √(x² + y²) ; arg z = arctan(y/x), adjusted by quadrant, −π < arg z ≤ π Polar / exponential
z = r(cosθ + i sinθ) = r e^(iθ) Operations
(z₁)(z₂): moduli multiply, arguments add z₁/z₂: moduli divide, arguments subtract Conjugate
z̄ = x − iy ; z z̄ = |z|² ; (z₁ z₂)* = z₁* z₂* De Moivre's Theorem
(cosθ + i sinθ)ⁿ = cos nθ + i sin nθ for n ∈ ℤ Use to derive multiple-angle identities, e.g. cos 3θ = 4cos³θ − 3cosθ.
Roots of Unity
n-th roots of unity: zₖ = e^(2πik/n), k = 0, 1, …, n−1 Sum of n-th roots of unity = 0 (n ≥ 2); they form a regular n-gon on the unit circle. Loci in the Argand Plane
|z − a| = r is a circle radius r centred at a |z − a| = |z − b| is the perpendicular bisector of ab arg(z − a) = θ is a half-line from a making angle θ with positive real axis Series, Induction & Polar Curves
Standard series, method of differences, induction proofs and polar areas.
Standard Summations
Σᵣ₌₁ⁿ r = n(n+1)/2 Σᵣ₌₁ⁿ r² = n(n+1)(2n+1)/6 Σᵣ₌₁ⁿ r³ = [n(n+1)/2]² = (Σr)² Method of Differences
If uᵣ = f(r) − f(r+1) (telescoping) then Σᵣ₌₁ⁿ uᵣ = f(1) − f(n+1) Express uᵣ in partial fractions to spot the telescoping structure.
Proof by Induction
1) Base case n = 1 (or smallest case) 2) Inductive step: assume true for n = k, prove for n = k+1 3) Conclude by induction Polar Curves r = f(θ)
Polar ↔ Cartesian
x = r cosθ, y = r sinθ ; r² = x² + y², tanθ = y/x Area of sector
A = ½ ∫_α^β r² dθ Common curves
r = a (circle), r = aθ (spiral), r = a(1 + cosθ) (cardioid), r = a cos kθ (rose) Hyperbolic Functions & Further Calculus
Hyperbolic identities, inverses and arc-length / surface-area integrals.
Definitions
sinh x = (eˣ − e⁻ˣ)/2 ; cosh x = (eˣ + e⁻ˣ)/2 ; tanh x = sinh x / cosh x Identities
cosh²x − sinh²x = 1 ; 1 − tanh²x = sech²x sinh 2x = 2 sinh x cosh x ; cosh 2x = cosh²x + sinh²x = 2cosh²x − 1 = 1 + 2sinh²x Derivatives & Inverses
d/dx (sinh x) = cosh x ; d/dx (cosh x) = sinh x ; d/dx (tanh x) = sech²x arsinh x = ln(x + √(x² + 1)) ; arcosh x = ln(x + √(x² − 1)), x ≥ 1 ; artanh x = ½ ln[(1+x)/(1−x)], |x| < 1 Arc Length & Surface Area
Arc length (Cartesian)
L = ∫ₐᵇ √(1 + (dy/dx)²) dx Arc length (parametric)
L = ∫_t₁^t₂ √((dx/dt)² + (dy/dt)²) dt Surface of revolution about x-axis
S = 2π ∫ₐᵇ y √(1 + (dy/dx)²) dx Vectors: Lines & Planes
3D geometry of lines, planes and shortest distances.
Plane Equations
Vector form
(r − a) · n = 0 Cartesian form
n₁x + n₂y + n₃z = d where d = a · n Through three points
n = (b − a) × (c − a) Cross Product
a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁) |a × b| = |a||b| sinθ ; perpendicular to both a and b Distances & Angles
Point to plane
d = |n₁x₀ + n₂y₀ + n₃z₀ − d| / |n| Skew lines
d = |(a₂ − a₁) · (b₁ × b₂)| / |b₁ × b₂| Angle between planes
cosθ = |n₁ · n₂| / (|n₁||n₂|) Further Mechanics
Circular motion, work–energy in 2D, elastic strings and SHM.
Circular Motion
v = rω ; centripetal acceleration a = v²/r = rω² Centripetal force F = mv²/r = mrω² For vertical circle: at top, T + mg = mv²/r ; at bottom, T − mg = mv²/r.
Hooke's Law & Elastic Strings
Tension
T = λx/l where λ = modulus of elasticity, l = natural length, x = extension Elastic potential energy
EPE = λx²/(2l) = ½kx² with k = λ/l Work–Energy in 2D
Work done by F along curve = ∫ F · dr Conservation of energy
KE₁ + PE₁ + EPE₁ + W_other = KE₂ + PE₂ + EPE₂ Simple Harmonic Motion
ẍ = −ω²x ; x = a cos(ωt + φ) or a sin(ωt + φ) Velocity & period
v² = ω²(a² − x²) ; T = 2π/ω ; vₘₐₓ = aω Further Statistics
Chi-squared, geometric, Poisson and continuous distributions.
Chi-Squared Test
χ² = Σ (Oᵢ − Eᵢ)² / Eᵢ Degrees of freedom
Goodness of fit: ν = (cells) − 1 − (parameters estimated) ; Contingency table: ν = (rows − 1)(cols − 1) Reject H₀ if χ²_calc > critical value at significance level α.
Geometric Distribution X ~ Geo(p)
P(X = r) = (1 − p)^(r−1) p, r = 1, 2, … Mean & variance
E(X) = 1/p ; Var(X) = (1 − p)/p² Poisson Distribution X ~ Po(λ)
P(X = r) = e^(−λ) λʳ / r! Mean & variance
E(X) = Var(X) = λ Sum of independent Poissons: X + Y ~ Po(λ_X + λ_Y).
Continuous Distributions
PDF condition
f(x) ≥ 0 and ∫_(−∞)^(∞) f(x) dx = 1 E(X) = ∫ x f(x) dx ; Var(X) = ∫ x² f(x) dx − [E(X)]² CDF
F(x) = P(X ≤ x) = ∫_(−∞)^x f(t) dt t-Distribution (Single Mean, σ Unknown)
t = (x̄ − μ₀)/(s/√n) with ν = n − 1 degrees of freedom How to Use This Formula Sheet
Boost your Cambridge exam confidence with these proven study strategies from our tutoring experts.
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Build on H240 First
Further Maths assumes total fluency with H240 pure. Make sure differentiation, integration, and trig identities are automatic before tackling further topics.
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Drill the Standard Techniques
Method of differences, induction proofs, and matrix transformations follow templates — drill the structure until the steps are reflexive.
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Visualise Geometric Topics
Sketch Argand diagrams, polar curves, and 3D vector geometry. Visualisation prevents algebraic mistakes and helps interpret answers.
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Practise the Optional Papers
Choose your option papers (mechanics or statistics) early and practise full past papers under timed conditions to build pacing.
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 Further Mathematics 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 Further Mathematics topics and equations does this formula sheet cover?
This page groups key Further Mathematics formulas in one place for revision. OCR A Level Further Maths A (H245) formula sheet for 2026: matrices, complex numbers, polar curves, hyperbolic functions, further mechanics (SHM, circular motion) and further statistics (chi-squared, Poisson). 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 Further 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 Further 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 Further Mathematics A (H245) for the 2026 exam series.
OCR provides a formula booklet in the exam — confirm with the latest specification which formulas are given and which must be memorised.