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A-Level Maths and Physics Key Formulae

Core formulae from pure maths, mechanics, statistics and physics that are common to AQA, Edexcel and OCR A-Level courses.

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Differentiating a power

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Differentiating a power

\frac{d}{d x} (a x^{n}) = an x^{n - 1}

Integrating a power

\int a x^{n} d x = \frac{a x ^{n + 1}}{n + 1} + c

for

n \neq = - 1

Chain rule

\frac{d y}{d x} = \frac{d y}{d u} \times \frac{d u}{d x}

Product rule

\frac{d}{d x} (uv) = u \frac{d v}{d x} + v \frac{d u}{d x}

Quotient rule

\frac{d}{d x} (\frac{u}{v}) = \frac{v \frac{d u}{d x} - u \frac{d v}{d x}}{v ^{2}}

Derivatives of

e^{k x}

and

ln x

\frac{d}{d x} e^{k x} = k e^{k x}

and

\frac{d}{d x} ln x = \frac{1}{x}

Quadratic formula

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

Discriminant: nature of roots

b^{2} - 4 a c > 0

: two real roots;

= 0

: one repeated root;

< 0

: no real roots

Sum of an arithmetic series

S_{n} = \frac{n}{2} [2 a + (n - 1) d]

Sum of a geometric series and sum to infinity

S_{n} = \frac{a ( 1 - r ^{n} )}{1 - r}

and

S_{\infty} = \frac{a}{1 - r}

for

∣ r ∣ < 1

Binomial expansion term

(a + b)^{n}

has general term

(r n) a^{n - r} b^{r}

Sine rule and cosine rule

\frac{a}{sin A} = \frac{b}{sin B}

and

c^{2} = a^{2} + b^{2} - 2 ab cos C

Arc length and sector area (radians)

s = r θ

and

A = \frac{1}{2} r^{2} θ

Key trigonometric identities

sin^{2} θ + cos^{2} θ \equiv 1

and

tan θ \equiv \frac{sin θ}{cos θ}

SUVAT equations

v = u + a t

s = u t + \frac{1}{2} a t^{2}

v^{2} = u^{2} + 2 a s

s = \frac{1}{2} (u + v) t

Normal distribution standardisation

Z = \frac{X - μ}{σ}

Binomial distribution

P (X = r) = (r n) p^{r} (1 - p)^{n - r}

, mean

n p

, variance

n p (1 - p)

Newton's second law and weight

F = ma

and

W = m g

Moment, work, power

Moment

= F d

; work

W = F s cos θ

; power

P = F v

Wave speed and the double-slit fringe spacing

v = f λ

and

w = \frac{λ D}{s}

Diffraction grating equation

d sin θ = nλ

Resistivity and resistance in a series/parallel

R = \frac{ρ L}{A}

; series

R = R_{1} + R_{2}

; parallel

\frac{1}{R} = \frac{1}{R _{1}} + \frac{1}{R _{2}}

EMF with internal resistance

ε = V + I r = I (R + r)

Young modulus

E = \frac{stress}{strain} = \frac{F / A}{Δ L / L}

Gravitational force and field strength

F = \frac{GM m}{r ^{2}}

and

g = \frac{GM}{r ^{2}}

Coulomb's law and uniform electric field

F = \frac{k Q _{1} Q _{2}}{r ^{2}}

with

k = \frac{1}{4 π ε _{0}}

;

E = \frac{V}{d}

Capacitor energy and discharge

E = \frac{1}{2} C V^{2}

;

Q = Q_{0} e^{- t / R C}

;

τ = R C

Force on a conductor and a moving charge

F = B I L sin θ

and

F = B q v sin θ

Faraday's law

ε = - N \frac{ΔΦ}{Δ t}

Circular motion

a = \frac{v ^{2}}{r} = ω^{2} r

and

F = \frac{m v ^{2}}{r}

;

ω = 2 π f

Simple harmonic motion

T = 2 π \frac{m}{k}

(spring),

T = 2 π \frac{l}{g}

(pendulum),

v_{ma x} = 2 π f A

Photon energy and photoelectric equation

E = h f = \frac{h c}{λ}

and

h f = ϕ + E_{k, ma x}

de Broglie wavelength

λ = \frac{h}{p} = \frac{h}{m v}

Radioactive decay

N = N_{0} e^{- λ t}

A = λ N

t_{1/2} = \frac{ln 2}{λ}

Ideal gas equations

p V = n R T = N k T

; mean KE per molecule

= \frac{3}{2} k T

A-Level Maths and Physics Key Formulae

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