All A-level printables

Further Maths formula sheet

Core pure and selected optional statistics and mechanics: 28 formula groups on 5 A4 pages, each labelled to show whether it is printed in the AQA and Edexcel exam booklets or must be learned. Includes conditions, common pitfalls and a learn-it checklist.

  • A-level Further Maths
  • 5 A4 pages
  • 14 to learn
  • Checked 11 October 2026
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PDF · 60 KB · prints at actual size on A4 · original revision aid, not an official exam booklet

First page of the Further Maths formula sheet with labelled formula boxes.
Page 1 preview

How the labels work

Further Maths builds on A-level Maths: revise the A-level Maths formula sheet too. Option content varies by board, so check which options you are taking.

Complex numbers

Cartesian and modulus–argument form

Learn: not in the booklets

z = x + iy = r(cos θ + i sin θ) = reiθ; r = |z| = √(x2 + y2); zz* = |z|2

Choose θ in the correct quadrant; principal argument −π < θ ≤ π.

Multiplication and division

Learn: not in the booklets

|z1z2| = |z1||z2|, arg(z1z2) = arg z1 + arg z2; |z1/z2| = |z1|/|z2|, arg(z1/z2) = arg z1 − arg z2

De Moivre’s theorem and roots of unity

In AQA and Edexcel booklets

[r(cos θ + i sin θ)]n = rn(cos nθ + i sin nθ)
zn = 1 has roots z = e2πki/n, k = 0, 1, …, n − 1

The n roots of zn = w are r1/n ei(θ + 2πk)/n, k = 0, …, n − 1 (learn this general form).

Loci in the Argand diagram

Learn: not in the booklets

|z − a| = r: circle, centre a, radius r; |z − a| = |z − b|: perpendicular bisector of a and b; arg(z − a) = θ: half-line from a

Matrices

2 × 2 determinant and inverse

Learn: not in the booklets

For M = (a b; c d): det M = ad − bc; M−1 = (1/(ad − bc)) (d −b; −c a)

M−1 exists only if det M ≠ 0. (AB)−1 = B−1A−1. |det M| is the area scale factor.

Transformation matrices

In AQA and Edexcel booklets

Rotation through θ anticlockwise about O: (cos θ −sin θ; sin θ cos θ)
Reflection in y = x tan θ: (cos 2θ sin 2θ; sin 2θ −cos 2θ)

Applying A then B is the single matrix BA. The AQA booklet also prints 3D rotations about the x-, y- and z-axes.

Invariant points and eigenvalues

Learn: not in the booklets

Invariant point: Mx = x. Eigenvalues: det(M − λI) = 0; eigenvectors satisfy (M − λI)v = 0, v ≠ 0

Eigenvalues are in AQA Further Maths content; check your specification.

Series, roots and proof

Standard sums

In AQA and Edexcel booklets

Σr=1n r2 = n(n + 1)(2n + 1)/6; Σr=1n r3 = n2(n + 1)2/4

Σr=1n r = n(n + 1)/2 is printed in the Edexcel booklet only.

Roots of polynomials

Learn: not in the booklets

ax2 + bx + c: α + β = −b/a, αβ = c/a
ax3 + bx2 + cx + d: Σα = −b/a, Σαβ = c/a, αβγ = −d/a

a ≠ 0. Use substitution (e.g. w = 2x + 1) to form equations with transformed roots.

Method of differences and proof by induction

Learn: not in the booklets

If ur = f(r) − f(r + 1), then Σr=1n ur = f(1) − f(n + 1).
Induction: prove the base case; assume true for n = k; prove for n = k + 1; conclude.

Maclaurin series

In AQA and Edexcel booklets

f(x) = f(0) + xf′(0) + x2f″(0)/2! + … + xrf(r)(0)/r! + …
ex = 1 + x + x2/2! + …; ln(1 + x) = x − x2/2 + x3/3 − … (−1 < x ≤ 1)
sin x = x − x3/3! + x5/5! − …; cos x = 1 − x2/2! + x4/4! − …

The Edexcel booklet also prints arctan x = x − x3/3 + x5/5 − … (−1 ≤ x ≤ 1).

Hyperbolic functions and further calculus

Hyperbolic definitions

Learn: not in the booklets

sinh x = (ex − e−x)/2; cosh x = (ex + e−x)/2; tanh x = sinh x / cosh x

Learn these: the booklets give identities, not the definitions.

Hyperbolic identities and inverses

In AQA and Edexcel booklets

cosh2x − sinh2x = 1
arsinh x = ln(x + √(x2 + 1)); arcosh x = ln(x + √(x2 − 1)) (x ≥ 1); artanh x = ½ ln((1 + x)/(1 − x)) (|x| < 1)

Inverse trigonometric and hyperbolic calculus

In AQA and Edexcel booklets

d/dx arcsin x = 1/√(1 − x2); d/dx arctan x = 1/(1 + x2); ∫1/(a2 + x2) dx = (1/a) arctan(x/a) + c; ∫1/√(a2 − x2) dx = arcsin(x/a) + c

Corresponding sinh, cosh and tanh results are printed too.

Volumes of revolution and mean value

Learn: not in the booklets

About the x-axis: V = π∫ab y2 dx; about the y-axis: V = π∫ x2 dy; mean value of f on [a, b] = (1/(b − a))∫ab f(x) dx

Polar coordinates

In AQA and Edexcel booklets

x = r cos θ, y = r sin θ; area = ½∫αβ r2 dθ

θ in radians. Only the area formula is printed; learn the conversions.

Arc length and surface area (option content)

In AQA and Edexcel booklets

s = ∫√(1 + (dy/dx)2) dx; S = 2π∫y√(1 + (dy/dx)2) dx (about the x-axis)

Parametric forms are printed too.

Vectors and differential equations

Scalar product and angles

Learn: not in the booklets

a·b = |a||b| cos θ = a1b1 + a2b2 + a3b3; perpendicular ⇔ a·b = 0

The booklets use a·b in other results but do not state this definition.

Vector product, lines and planes

In AQA and Edexcel booklets

a × b = |a||b| sin θ n̂; line r = a + λb; plane r = a + λb + μc or r·n = a·n

Cartesian plane: n1x + n2y + n3z = d.

First-order linear equations

Learn: not in the booklets

dy/dx + P(x)y = Q(x): integrating factor I = e∫P dx; then d/dx (Iy) = IQ

Divide first so the coefficient of dy/dx is 1.

Second-order linear equations

Learn: not in the booklets

ay″ + by′ + cy = 0, auxiliary equation am2 + bm + c = 0:
distinct real m1, m2: y = Aem₁x + Bem₂x; repeated m: y = (A + Bx)emx; m = p ± qi: y = epx(A cos qx + B sin qx)

For f(x) ≠ 0 add a particular integral to the complementary function.

Simple harmonic motion

Learn: not in the booklets

ẍ = −ω2x ⇒ x = A cos ωt + B sin ωt; period T = 2π/ω; v2 = ω2(a2 − x2)

a is the amplitude.

Numerical methods

AQA booklet only

Euler: yn+1 = yn + h f(xn, yn); Simpson’s rule (n even): ∫ab y dx ≈ (h/3)[y0 + 4(y1 + y3 + …) + 2(y2 + y4 + …) + yn]; mid-ordinate rule

Not in the Edexcel booklet.

Further statistics and mechanics (options)

Poisson distribution

In AQA and Edexcel booklets

X ~ Po(λ): P(X = x) = e−λλx/x!; E(X) = Var(X) = λ

Events occur singly, independently and at a constant average rate.

Chi-squared test

In AQA and Edexcel booklets

X2 = Σ(O − E)2/E; degrees of freedom: (rows − 1)(columns − 1) for contingency tables

Combine classes so each expected frequency is at least 5.

Work, energy and power

Learn: not in the booklets

Work = Fs cos θ; KE = ½mv2; GPE = mgh; power P = Fv

Work–energy principle: work done by non-conservative forces = change in mechanical energy.

Circular motion

Edexcel booklet only

v = rω; acceleration towards centre = v2/r = rω2

Printed in the Edexcel booklet; AQA students must learn it.

Centres of mass

In AQA and Edexcel booklets

x̄ = Σmx/Σm; triangular lamina: ⅔ along median from vertex; solid hemisphere: 3r/8 from centre; solid cone: h/4 from base

Formulae to learn

These are not printed in the AQA or Edexcel booklets. The last page of the PDF has the same list as tick-boxes.

Official sources

Checked on 11 October 2026. Booklet versions are the ones the boards list as current; use the copy your board supplies in the exam.

Use the formulae in practice

Start A-level Daily 10 Maths →
Five pure, three mechanics and two statistics questions every day.
A-level Maths formula sheetA-level maths worksheetA-level maths pathwayIntegration guide

Common questions

Is this the official exam formulae booklet?

No. It is an original MathsDaily revision sheet. In the exam you use the booklet your board supplies; this sheet shows which formulae are in that booklet and which you must learn.

Which formulae do I need to learn?

14 of the 28 formula groups here are labelled “Learn: not in the booklets”. The final page of the PDF is a tick-list of them.

Does it cover OCR?

OCR prints its formulae on the inside pages of each question paper rather than in a separate booklet, so OCR students should check the list in a recent paper. Most of the “learn” formulae are the same.