A-Level Maths Formula Sheet

The A-Level Maths curriculum introduces advanced mathematical concepts that form the foundation for many STEM degrees and careers. To succeed, you need to be comfortable with a wide range of complex formulas. This A-Level Maths formula sheet brings together the critical equations and identities you must know, spanning across algebra, calculus, trigonometry, logarithms, vectors, and probability. From integration and differentiation rules to trigonometric identities and vector equations, this sheet serves as a quick reference guide to support your studies. While some formulas are provided in the exam booklet, knowing them by heart will save you valuable time and reduce errors. Use this resource to test your memory, reinforce your understanding during homework, and streamline your exam revision process for maximum efficiency and success.

algebra

Quadratic Formula

Class 10algebra

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

Roots of a quadratic equation ax^2 + bx + c = 0

#roots#equations#quadratic

Difference of Squares

Class 10algebra

a^2 - b^2 = (a - b)(a + b)

Factoring difference of two squares

#factoring#polynomials

Sum of Cubes

Class 10algebra

a^3 + b^3 = (a + b)(a^2 - ab + b^2)

Factoring sum of two cubes

#factoring#cubes

Difference of Cubes

Class 10algebra

a^3 - b^3 = (a - b)(a^2 + ab + b^2)

Factoring difference of two cubes

#factoring#cubes

Binomial Theorem

Class 12algebra

(x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k

Expansion of a binomial power

#expansion#series#binomial

Arithmetic Progression (nth Term)

Class 10algebra

a_n = a_1 + (n-1)d

nth term of an AP with common difference d

#sequences#ap

Arithmetic Progression (Sum)

Class 10algebra

S_n = \frac{n}{2}(2a_1 + (n-1)d) = \frac{n}{2}(a_1 + a_n)

Sum of first n terms of an AP

#series#ap

Geometric Progression (nth Term)

Class 12algebra

a_n = a_1 r^{n-1}

nth term of a GP with common ratio r

#sequences#gp

Geometric Progression (Sum)

Class 12algebra

S_n = \frac{a_1(1-r^n)}{1-r} \quad (r \neq 1)

Sum of first n terms of a GP

#series#gp

Infinite Geometric Series

Class 12algebra

S_\infty = \frac{a_1}{1-r} \quad (|r| < 1)

Sum of infinite GP when ratio is strictly between -1 and 1

#series#infinite

Vieta's Formulas (Quadratic)

Class 10algebra

x_1 + x_2 = -\frac{b}{a}, \quad x_1 x_2 = \frac{c}{a}

Sum and product of roots of ax^2 + bx + c = 0

#roots#vieta#polynomials

Completing the Square

Class 10algebra

ax^2 + bx + c = a\left(x + \frac{b}{2a}\right)^2 + \left(c - \frac{b^2}{4a}\right)

Rewriting quadratic into vertex form

#quadratic#factoring

calculus

Power Rule (Derivative)

Class 12calculus

\frac{d}{dx}[x^n] = n x^{n-1}

Derivative of a power function

#derivative#power

Product Rule

Class 12calculus

\frac{d}{dx}[u \cdot v] = u'v + uv'

Derivative of product of two functions

#derivative#product

Quotient Rule

Class 12calculus

\frac{d}{dx}\left[\frac{u}{v}\right] = \frac{u'v - uv'}{v^2}

Derivative of quotient of two functions

#derivative#quotient

Chain Rule

Class 12calculus

\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)

Derivative of composite functions

#derivative#chain

Derivative of Exponential

Class 12calculus

\frac{d}{dx}[e^x] = e^x, \quad \frac{d}{dx}[a^x] = a^x \ln a

Derivative of natural and base-a exponential functions

#derivative#exponential

Derivative of Logarithm

Class 12calculus

\frac{d}{dx}[\ln x] = \frac{1}{x}, \quad \frac{d}{dx}[\log_a x] = \frac{1}{x \ln a}

Derivative of natural and base-a logarithms

#derivative#logarithm

Standard Trig Derivatives

Class 12calculus

\frac{d}{dx}[\sin x] = \cos x, \quad \frac{d}{dx}[\cos x] = -\sin x, \quad \frac{d}{dx}[\tan x] = \sec^2 x

Derivatives of basic trigonometric functions

#derivative#trigonometry

Power Rule (Integral)

Class 12calculus

\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)

Indefinite integral of power function

#integral#power

Reciprocal Integral

Class 12calculus

\int \frac{1}{x} \, dx = \ln|x| + C

Integral of 1/x yielding natural logarithm

#integral#logarithm

Integration by Parts

Higher Edcalculus

\int u \, dv = uv - \int v \, du

Standard integration technique for products

#integral#parts

Fundamental Theorem of Calculus

Higher Edcalculus

\int_a^b f(x) \, dx = F(b) - F(a)

Evaluating definite integral from antiderivative F(x)

#integral#theorem

L'Hôpital's Rule

Higher Edcalculus

\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}

Evaluating indeterminate forms (0/0 or ∞/∞)

#limit#lhopital

trigonometry

Pythagorean Identity

Class 10trigonometry

\sin^2\theta + \cos^2\theta = 1

Fundamental trigonometric identity relating sine and cosine

#identities#pythagorean

Secant & Tangent Identity

Class 10trigonometry

1 + \tan^2\theta = \sec^2\theta

Pythagorean identity for tangent and secant

#identities#pythagorean

Cosecant & Cotangent Identity

Class 10trigonometry

1 + \cot^2\theta = \csc^2\theta

Pythagorean identity for cotangent and cosecant

#identities#pythagorean

Sine Addition & Subtraction

Class 12trigonometry

\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B

Sine of sum or difference of two angles

#compound#addition

Cosine Addition & Subtraction

Class 12trigonometry

\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B

Cosine of sum or difference of two angles

#compound#addition

Tangent Addition & Subtraction

Class 12trigonometry

\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}

Tangent of sum or difference of two angles

#compound#addition

Double Angle (Sine)

Class 12trigonometry

\sin(2\theta) = 2\sin\theta\cos\theta

Sine of double angle

#double-angle

Double Angle (Cosine)

Class 12trigonometry

\cos(2\theta) = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta

Cosine of double angle formulas

#double-angle

Half Angle Formulas

Class 12trigonometry

\sin\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1-\cos\theta}{2}}, \quad \cos\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1+\cos\theta}{2}}

Sine and cosine of half angle

#half-angle

Law of Sines

Class 10trigonometry

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R

Relates sides of a triangle to the sines of its angles

Interactive Triangle Solver (SAS):

Side a

Side b

Angle C (°)

Side c (opposite to ∠C):—

#triangles#sines

Law of Cosines

Class 10trigonometry

c^2 = a^2 + b^2 - 2ab\cos C

Relates sides of a triangle to cosine of an angle

Interactive Triangle Solver (SAS):

Side a

Side b

Angle C (°)

Side c (opposite to ∠C):—

#triangles#cosines

logarithms

Product Rule for Logs

Class 10logarithms

\log_b(xy) = \log_b(x) + \log_b(y)

Logarithm of a product equals sum of logarithms

#product#logarithm

Quotient Rule for Logs

Class 10logarithms

\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)

Logarithm of a quotient equals difference of logarithms

#quotient#logarithm

Power Rule for Logs

Class 10logarithms

\log_b(x^k) = k \log_b(x)

Exponent comes out as a multiplier

#power#exponent#logarithm

Change of Base Formula

Class 12logarithms

\log_b(x) = \frac{\log_k(x)}{\log_k(b)} = \frac{\ln x}{\ln b} = \frac{\log_{10} x}{\log_{10} b}

Converting logarithm to a different base k

#base#conversion

Identity & Zero Properties

Class 10logarithms

\log_b(b) = 1, \quad \log_b(1) = 0

Basic identity properties of logarithms

#identity#logarithm

Inverse Properties

Class 12logarithms

b^{\log_b x} = x, \quad \log_b(b^x) = x

Logarithms and exponentials are inverse functions

#inverse#exponential

Natural Log & Euler Constant

Higher Edlogarithms

\ln(x) = \log_e(x), \quad e = \lim_{n \to \infty}\left(1 + \frac{1}{n}\right)^n \approx 2.71828

Definition of natural logarithm and base e

#natural#euler

Exponential Laws

Class 10logarithms

a^m \cdot a^n = a^{m+n}, \quad \frac{a^m}{a^n} = a^{m-n}, \quad (a^m)^n = a^{mn}

Fundamental laws of exponent arithmetic

#exponential#powers

vectors

Vector Dot Product

Class 12vectors

\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta = a_x b_x + a_y b_y + a_z b_z

Scalar product of two vectors

#vectors#dot-product

Vector Cross Product

Class 12vectors

\vec{a} \times \vec{b} = |\vec{a}||\vec{b}|\sin\theta \, \hat{n} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_x & a_y & a_z \\ b_x & b_y & b_z \end{vmatrix}

Vector product yielding orthogonal vector

#vectors#cross-product

Vector Magnitude

Class 12vectors

|\vec{a}| = \sqrt{a_x^2 + a_y^2 + a_z^2}

Euclidean norm or length of a vector

#vectors#magnitude

Unit Vector

Class 12vectors

\hat{a} = \frac{\vec{a}}{|\vec{a}|}

Vector of length 1 pointing in same direction

#vectors#unit-vector

Scalar Projection of A onto B

Class 12vectors

\text{proj}_{\vec{b}}(\vec{a}) = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}

Magnitude of vector a in direction of vector b

#vectors#projection

Matrix Determinant (2x2)

Class 12vectors

\det\begin{bmatrix} a & b \\ c & d \end{bmatrix} = ad - bc

Determinant of a 2 by 2 matrix

#matrices#determinant

Matrix Inverse (2x2)

Class 12vectors

\begin{bmatrix} a & b \\ c & d \end{bmatrix}^{-1} = \frac{1}{ad-bc}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}

Inverse of an invertible 2 by 2 matrix

#matrices#inverse

Characteristic Equation (Eigenvalues)

Higher Edvectors

\det(A - \lambda I) = 0

Equation for finding eigenvalues λ of matrix A

#matrices#eigenvalues

probability

Permutations Formula

Class 10probability

P(n, r) = {}^n P_r = \frac{n!}{(n-r)!}

Number of ordered arrangements of r elements from n

#combinatorics#permutations

Combinations Formula

Class 10probability

C(n, r) = {}^n C_r = \binom{n}{r} = \frac{n!}{r!(n-r)!}

Number of unordered selections of r elements from n

#combinatorics#combinations

Addition Rule of Probability

Class 10probability

P(A \cup B) = P(A) + P(B) - P(A \cap B)

Probability of union of two events

#probability#union

Multiplication Rule

Class 12probability

P(A \cap B) = P(A) \cdot P(B|A)

Probability of simultaneous occurrence of two events

#probability#intersection

Conditional Probability

Class 12probability

P(A|B) = \frac{P(A \cap B)}{P(B)} \quad (P(B) > 0)

Probability of event A occurring given event B has occurred

#probability#conditional

Bayes' Theorem

Higher Edprobability

P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} = \frac{P(B|A) P(A)}{\sum P(B|A_i) P(A_i)}

Calculates posterior probability based on prior knowledge

#probability#bayes

Binomial Distribution PMF

Class 12probability

P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}

Probability of k successes in n independent Bernoulli trials

#distribution#binomial

Poisson Distribution PMF

Higher Edprobability

P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}

Probability of k events in a fixed interval with average rate λ

#distribution#poisson

Expected Value (Discrete)

Class 12probability

E[X] = \sum_{i} x_i \cdot P(X = x_i)

Weighted average of all possible values of discrete random variable

#statistics#mean#expectation

Variance & Standard Deviation

Class 12probability

\text{Var}(X) = E[X^2] - (E[X])^2, \quad \sigma = \sqrt{\text{Var}(X)}

Measure of dispersion of a random variable

#statistics#variance

Standard Normal Distribution (PDF)

Higher Edprobability

f(z) = \frac{1}{\sqrt{2\pi}} e^{-z^2/2}, \quad z = \frac{x - \mu}{\sigma}

Probability density function of standard normal variable

#distribution#normal#z-score

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