Formulae and identities GCSE/A-level - Essential math summary - Physics and calculus included

Oct 20, 2026

Uploaded by Noah

A compact revision sheet of essential GCSE and A-level mathematics formulae and identities, organized by major topics. It covers coordinate geometry and straight-line equations, arithmetic and geometric sequences, logarithm and index laws, kinematics with variable acceleration, vector magnitude, and core trigonometric identities. It also includes statistics (mean and standard normal), mensuration for circle sector arc length and area, and calculus with differentiation and integration rules, plus forces and equilibrium using weight, friction, and Newton’s second law.

LanguageEnglish

CategoryExam

FormatPDF

Pages1

Reading Time3 min

AccessFree to download

Coordinate geometry

Sequences

Laws of logarithms

Kinematics

Vectors

Trigonometry

Laws of indices

Statistics

Mensuration

Calculus and differential equations

Differentiation

Integration

Forces and equilibrium

Formulae and identities you’ll need to know Remember: you'll still need your GCSE formulaeat A-level Coordinate geometry A straight line graph, gradient m passing through (x1, y1 ) has equation y-y1 = m (x-x1 ) Straight lines with gradients m1 and m2 are perpendicular when m1 m 2 = -1 Sequences General term of an arithmetic progression: un = a + ( n-1)d General term of a geometric progression: u = arn-1 n Laws of Logarithms x = an ‹=› n = log ax for a> 0 and x> 0 log ax + logay---loga (xy) log ax-logay---loga ~~(~~yx~~) ~~klogax---loga (xk) Kinematics For motion in a straight line with variable acceleration: v = rtr = ∫vdt a = = dd v = ∫ad t Vectors |xi+yj+zk|=√ (x2 +y2 +z2 ) Trigonometry cos 2A + sin 2A---1 sec 2A---1 + tan 2Acosec 2A---1 + cot 2Asin2A---2sinAcosA cos2A---cos 2A-sin2Atan2A---~~1 2~~-t~~ ~~atnaAn~~2~~A Laws of Indices ax ay---ax+y ax ÷ ay---ax-y ( ax ) y---axy Statistics The mean of a set of data: x = Σnx = Σ~~ƒ~~xΣƒ The standard Normal variable: Z = X- µσ where X ~ N (µ,σ2 ) Mensuration For a circle of radius r, where an angle at the centre of Θ radians subtendsan arc of length s and encloses an associated sector of area A: s = rΘ A = ½ r2 Θ Calculus and Differential Equations Differentiation | Function | Derivative | | --- | --- | | xn | nx n-1 | | sinkx | k coskx | | coskx | -ksinkx | | ekx | kekx | | Inx | 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 | | --- | --- | | xn | n~~ ~~~~1~~xn+1 + c, n≠ -1 | | coskx | ~~1k~~sinkx + c | | sinkx | -~~1k~~ coskx + c | | ekx | 1 kx ke + c | | 1 x | In|x | + c, x ≠0 | | f'(x ) + g'(x ) | f(x ) + g (x ) + c | | f'( g (x )) g'(x ) | f( g (x )) + c | Area under a curve: = y dx (y ≥ 0) a Forces and Equilibrium Weight = mass x g Friction: F ≤ µR Newton's second law in the form: F = ma

FAQs

Which topics of GCSE/A-level maths are summarized in this document?

It covers coordinate geometry, sequences, logarithm and index laws, kinematics, vectors, trigonometry, statistics, mensuration, calculus (differentiation and integration), and forces/equilibrium.

How is the perpendicular condition for two straight lines stated?

Two straight lines with gradients m1 and m2 are perpendicular when m1·m2 = -1.

What differentiation and integration rules are included?

The sheet lists derivative rules for powers, trig functions, exponential and logarithmic forms, and sum/difference/product/chain rules, plus corresponding basic integration results and the area-under-a-curve expression.

Oct 20, 2026

Uploaded by Noah