A Levels Maths Formula Sheet

A Level Maths formula sheet

Algebra and Functions

Laws of Indices:

$\textbf{1.} \ a^m \cdot a^n = a^{m+n}$

$\textbf{2.} \frac{a^m}{a^n} = a^{m-n}$

$\textbf{3.} \ (a^m)^n = a^{mn}$

$\textbf{4.} \ a^0 = 1$

$\textbf{5.} \ a^{-n} = \frac{1}{a^n}$

Quadratic Formula:

$\textbf{1.} \ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Completing the Square:

$\textbf{1.} \ ax^2 + bx + c = a\left(x + \frac{b}{2a}\right)^2 - \frac{b^2 - 4ac}{4a}$

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Coordinate Geometry

Distance Between Two Points:

$\textbf{1.} \ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Midpoint of a Line Segment:

$\textbf{1.} \ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

Midpoint of a Line Segment:

$\textbf{1.} \ m = \frac{y_2 - y_1}{x_2 - x_1}$

Trigonometry

Basic Identities:

$\textbf{1.} \ \sin^2 \theta + \cos^2 \theta = 1$

$\textbf{2.} \ \tan \theta = \frac{\sin \theta}{\cos \theta}$

Sine Rule:

$\textbf{1.} \ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

Cosine Rule:

$\textbf{1.} \ a^2 = b^2 + c^2 - 2bc \cos A$

Area of a Triangle:

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

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Differentiation

Basic Rules:

$\textbf{1.} \ \frac{d}{dx}(x^n) = nx^{n-1}$

$\textbf{2.} \ \frac{d}{dx}(\sin x) = \cos x$

$\textbf{3.} \ \frac{d}{dx}(\cos x) = -\sin x$

$\textbf{4.} \ \frac{d}{dx}(\tan x) = \sec^2 x$

Product Rule:

$\textbf{1.} \ \frac{d}{dx}(uv) = u'v + uv'$

Quotient Rule:

$\textbf{1.} \ \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2}$

chain Rule:

$\textbf{1.} \ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$

Integration

Basic Rules:

$\textbf{1.} \ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad \text{for } n \ne -1$

$\textbf{2.} \ \int \frac{1}{x} \, dx = \ln |x| + C$

Standard Integrals:

$\textbf{1.} \ \int \sin x \, dx = -\cos x + C$

$\textbf{1.} \ \int \cos x \, dx = \sin x + C$

Statistics

Mean of Data:

$\textbf{1.} \ \bar{x} = \frac{\sum x}{n}$

Variance:

$\textbf{1.} \text{Variance} = \frac{\sum (x - \bar{x})^2}{n}$

Standard Deviation:

$\textbf{1.} \ \text{Standard Deviation} = \sqrt{\text{Variance}}$

Probability Rules:

$\textbf{1.} \ P(A \cup B) = P(A) + P(B) - P(A \cap B)$

$\textbf{1.} \ P(A \mid B) = \frac{P(A \cap B)}{P(B)}$

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