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AQA A Level Maths Formula Sheet

Every formula in the AQA A Level Mathematics (7357) formulae booklet, plus every formula students must learn by heart for pure maths, statistics and mechanics. Read it here or download the PDF.

At a Glance

AQA A Level Mathematics

A formulae booklet is provided in every paper, and a calculator is allowed. This page shows exactly what is on it, and the formulas that are not given, which students most often forget.

Given in the Exam21 formulas in the booklet
Learn by Heart21 formulas to learn, the same for every board
Read it online ↓AQA A Level revision checklist →
Formula Sheet

AQA A Level Maths Formulas

Everything in the formulae booklet, grouped into pure maths, mechanics and statistics, then the formulas you must learn.

1Given in Your Exam

These formulas are in the AQA "Formulae and Statistical Tables" booklet given out in every paper. You don't need to memorise them, but learn when and how to use each one. It also includes statistical tables for the binomial and normal distributions, although most students use their calculator for these.

Pure

Binomial Series
(a + b)n = an + (n1)an−1b + … + (nr)an−rbr + … + bn
(nr) = n!r!(n − r)!
For positive whole numbers n
General Binomial Expansion
(1 + x)n = 1 + nx + n(n − 1)2!x2 + … + n(n − 1)…(n − r + 1)r!xr + …
Valid for |x| < 1 when n is negative or a fraction
Arithmetic Series
Sn = 12n(a + l) = 12n[2a + (n − 1)d]
a is the first term, l the last term, d the common difference
Geometric Series
Sn = a(1 − rn)1 − r
S∞ = a1 − r for |r| < 1
a is the first term, r the common ratio
Small Angle Approximations
sin θ ≈ θ
cos θ ≈ 1 − θ22
tan θ ≈ θ
θ in radians
Compound Angle Formulae
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 B1 ∓ tan A tan B
Derivatives of Trig Functions
tan x → sec2 x
sec x → sec x tan x
cot x → −cosec2 x
cosec x → −cosec x cot x
f(x) → f′(x)
Product and Chain Rules
ddx[f(x)g(x)] = f′(x)g(x) + f(x)g′(x)
ddx[f(g(x))] = f′(g(x))g′(x)
Quotient Rule
y = uv ⇒ dydx = vdudx − udvdxv2
Differentiation From First Principles
f′(x) = limh→0 f(x + h) − f(x)h
Integration by Parts
∫udvdx dx = uv − ∫vdudx dx
Integrating f′(x) ÷ f(x)
∫f′(x)f(x) dx = ln|f(x)| + c
Integrals of tan x and cot x
∫tan x dx = ln|sec x| + c
∫cot x dx = ln|sin x| + c
Newton–Raphson Method
xn+1 = xn − f(xn)f′(xn)
The Trapezium Rule
∫ab y dx ≈ h2{(y0 + yn) + 2(y1 + … + yn−1)}
h = b − an

Mechanics

Constant Acceleration (suvat)
v = u + at
s = ut + 12at2
s = vt − 12at2
s = 12(u + v)t
v2 = u2 + 2as
Motion in a straight line with constant acceleration

Statistics

Probability: A or B
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Probability: A and B
P(A ∩ B) = P(A) × P(B|A)
Standard Deviation
σ = √Σ(x − x̄)2n = √Σx2n − x̄2
The Binomial Distribution
If X ~ B(n, p):
P(X = x) = (nx)px(1 − p)n−x
Mean = np
Variance = np(1 − p)
Sample Mean From a Normal Distribution
X̄ ~ N(μ, σ2n)
X̄ − μσ/√n ~ N(0, 1)
Used in hypothesis tests for the mean

2Learn These by Heart

These are not given in the exam. Students lose marks on them every year.

Algebra and Functions

The Quadratic Formula
x = −b ± √b2 − 4ac2a
Laws of Indices
ax × ay = ax+y
ax ÷ ay = ax−y
(ax)y = axy
Laws of Logarithms
x = an ⇔ n = loga x
loga x + loga y = loga(xy)
loga x − loga y = logaxy
k loga x = loga(xk)
a > 0, x > 0
Straight Lines
y − y1 = m(x − x1)
Perpendicular: m1m2 = −1
nth Terms of Sequences
Arithmetic: un = a + (n − 1)d
Geometric: un = arn−1

Trigonometry

Sine Rule, Cosine Rule and Area
asin A = bsin B = csin C
a2 = b2 + c2 − 2bc cos A
Area = 12ab sin C
Pythagorean Identities
cos2A + sin2A ≡ 1
sec2A ≡ 1 + tan2A
cosec2A ≡ 1 + cot2A
Double Angle Formulae
sin 2A ≡ 2 sin A cos A
cos 2A ≡ cos2A − sin2A
tan 2A ≡ 2 tan A1 − tan2A
cos 2A can also be written as 2cos²A − 1 or 1 − 2sin²A
Arc Length and Sector Area
s = rθ
A = 12r2θ
θ in radians

Mensuration

Circles
C = 2πr = πd
A = πr2
Pythagoras' Theorem
c2 = a2 + b2
c is the hypotenuse
Trapezium and Prism
Area of a trapezium = 12(a + b)h
Volume of a prism = area of cross-section × length

Calculus and Vectors

Standard Derivatives
xn → nxn−1
sin kx → k cos kx
cos kx → −k sin kx
ekx → kekx
ln x → 1x
f(x) → f′(x)
Product and Chain Rules
(fg)′ = f′g + fg′
ddxf(g(x)) = f′(g(x))g′(x)
Standard Integrals
xn → xn+1n + 1 (n ≠ −1)
cos kx → 1ksin kx
sin kx → −1kcos kx
ekx → 1kekx
1x → ln|x|
Add + c to each
Reverse Chain Rule and Area
∫f′(g(x))g′(x) dx = f(g(x)) + c
Area under a curve = ∫ab y dx (y ≥ 0)
Magnitude of a Vector
|xi + yj + zk| = √x2 + y2 + z2

Mechanics

Forces and Newton's Second Law
Weight = mg
Friction: F ≤ μR
F = ma
Use g = 9.8 m s⁻² unless told otherwise
Variable Acceleration
v = drdt
a = dvdt = d2rdt2
r = ∫v dt
v = ∫a dt

Statistics

Mean
x̄ = Σxn = ΣfxΣf
Standardising a Normal Variable
Z = X − μσ, where X ~ N(μ, σ2)

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