Formulae that students are expected to know for A Level Mathematics are given below and will not appear in the booklet Mathematical Formulae and Statistical Tables, which will be provided for use with the paper.

Pure Mathematics Algebra, Geometry & Sequences

Quadratic Equations

Roots of quadratic equations

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

Laws of Indices

Multiplication Law

a^x a^y = a^{x+y}

Division Law

a^x \div a^y = a^{x-y}

Power of a Power Law

(a^x)^y = a^{xy}

Laws of Logarithms

Definition

x = a^n \iff n = \log_a x \quad \text{for } a > 0 \text{ and } x > 0

Addition Law

\log_a x + \log_a y \equiv \log_a(xy)

Subtraction Law

\log_a x - \log_a y \equiv \log_a\left(\frac{x}{y}\right)

Power Law

k \log_a x \equiv \log_a(x^k)

Coordinate Geometry

Equation of a straight line

y - y_1 = m(x - x_1) \quad \text{with gradient } m \text{ passing through } (x_1, y_1)

Perpendicular lines condition

m_1 m_2 = -1

Sequences

General term of an arithmetic progression

u_n = a + (n-1)d

General term of a geometric progression

u_n = a r^{n-1}

Trigonometry & Mensuration Identities and Rules

Trigonometry in Triangle $ABC$

Sine rule

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

Cosine rule

a^2 = b^2 + c^2 - 2bc \cos A

Area of a triangle

\text{Area} = \frac{1}{2}ab \sin C

Trigonometric Identities

Pythagorean Identity (Sine/Cosine)

\cos^2 A + \sin^2 A \equiv 1

Pythagorean Identity (Secant/Tangent)

\sec^2 A \equiv 1 + \tan^2 A

Pythagorean Identity (Cosecant/Cotangent)

\csc^2 A \equiv 1 + \cot^2 A

Double Angle (Sine)

\sin 2A \equiv 2 \sin A \cos A

Double Angle (Cosine)

\cos 2A \equiv \cos^2 A - \sin^2 A

Double Angle (Tangent)

\tan 2A \equiv \frac{2 \tan A}{1 - \tan^2 A}

Mensuration

Circumference and area of circle

C = 2\pi r = \pi d, \quad A = \pi r^2

Pythagoras' theorem

c^2 = a^2 + b^2

Area of a trapezium

\text{Area} = \frac{1}{2}(a+b)h

Volume of a prism

\text{Volume} = \text{area of cross section} \times \text{length}

Arc length of sector (radians)

s = r\theta

Area of sector (radians)

A = \frac{1}{2}r^2\theta

Calculus & Differential Equations Differentiation & Integration Rules

Differentiation

Function	Derivative
$x^n$	$n x^{n-1}$
$\sin kx$	$k \cos kx$
$\cos kx$	$-k \sin kx$
$e^{kx}$	$k e^{kx}$
$\ln x$	$\frac{1}{x}$
$f(x) + g(x)$	$f'(x) + g'(x)$
$f(x)g(x)$	$f'(x)g(x) + f(x)g'(x)$
$f(g(x))$	$f'(g(x))g'(x)$

Integration

Function	Integral
$x^n$	$\frac{1}{n+1} x^{n+1} + c \quad (n \neq -1)$
$\cos kx$	$\frac{1}{k} \sin kx + c$
$\sin kx$	$-\frac{1}{k} \cos kx + c$
$e^{kx}$	$\frac{1}{k} e^{kx} + c$
$\frac{1}{x}$	$\ln\|x\| + c \quad (x \neq 0)$
$f'(x) + g'(x)$	$f(x) + g(x) + c$
$f'(g(x))g'(x)$	$f(g(x)) + c$
Area under a curve	$\int_{a}^{b} y\,dx \quad (y \ge 0)$

Vectors, Statistics & Mechanics Applications

Vectors

Magnitude of 3D vector

|x\mathbf{i} + y\mathbf{j} + z\mathbf{k}| = \sqrt{x^2 + y^2 + z^2}

Statistics

Mean of a set of data

\bar{x} = \frac{\sum x}{n} = \frac{\sum fx}{\sum f}

Standard Normal variable

Z = \frac{X - \mu}{\sigma} \quad \text{where } X \sim \text{N}(\mu, \sigma^2)

Mechanics - Forces and Equilibrium

Weight

W = m g

Friction

F \le \mu R

Newton's second law

F = ma

Mechanics - Kinematics (Variable Acceleration)

Velocity from displacement

v = \frac{dr}{dt}

Acceleration from velocity

a = \frac{dv}{dt} = \frac{d^2r}{dt^2}

Displacement from velocity

r = \int v\,dt

Velocity from acceleration

v = \int a\,dt

Mathematical Formulae - A-Levels Maths