Quadratic Formula

For \( ax^2 + bx + c = 0 \):

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

Discriminant

\[ \Delta = b^2 - 4ac \]

• \( \Delta > 0 \): Two distinct real roots

• \( \Delta = 0 \): One repeated root

• \( \Delta < 0 \): No real roots (two complex roots)

Binomial Expansion

\[ (a + b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r \]

Where \( \binom{n}{r} = \frac{n!}{r!(n-r)!} = {}^nC_r \)

For \( |x| < 1 \) and \( n \in \mathbb{R} \):

\[ (1 + x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \cdots \]

Laws of Indices

\[ a^m \times a^n = a^{m+n} \]

\[ \frac{a^m}{a^n} = a^{m-n} \]

\[ (a^m)^n = a^{mn} \]

\[ a^0 = 1 \quad (a \neq 0) \]

\[ a^{-n} = \frac{1}{a^n} \]

\[ a^{\frac{1}{n}} = \sqrt[n]{a} \]

Laws of Logarithms

\[ \log_a(xy) = \log_a x + \log_a y \]

\[ \log_a\left(\frac{x}{y}\right) = \log_a x - \log_a y \]

\[ \log_a(x^k) = k \log_a x \]

\[ \log_a a = 1 \]

\[ \log_a 1 = 0 \]

\[ a^{\log_a x} = x \]

\[ \log_a x = \frac{\log_b x}{\log_b a} \]

Arithmetic Sequence

General term: \( u_n = a + (n-1)d \)

Sum of first n terms:

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

Where \( a \) = first term, \( d \) = common difference, \( l \) = last term

Geometric Sequence

General term: \( u_n = ar^{n-1} \)

Sum of first n terms:

\[ S_n = \frac{a(1-r^n)}{1-r} = \frac{a(r^n-1)}{r-1} \quad (r \neq 1) \]

Sum to infinity:

\[ S_\infty = \frac{a}{1-r} \quad (|r| < 1) \]

Standard Derivatives

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

\[ \frac{d}{dx}(e^x) = e^x \]

\[ \frac{d}{dx}(e^{kx}) = ke^{kx} \]

\[ \frac{d}{dx}(\ln x) = \frac{1}{x} \]

\[ \frac{d}{dx}(\sin x) = \cos x \]

\[ \frac{d}{dx}(\cos x) = -\sin x \]

\[ \frac{d}{dx}(\tan x) = \sec^2 x \]

\[ \frac{d}{dx}(\sin kx) = k\cos kx \]

\[ \frac{d}{dx}(\cos kx) = -k\sin kx \]

Differentiation Rules

Product Rule:

\[ \frac{d}{dx}[u \cdot v] = u\frac{dv}{dx} + v\frac{du}{dx} \]

Quotient Rule:

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

Chain Rule:

\[ \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} \]

Stationary Points

At stationary points: \( \frac{dy}{dx} = 0 \)

Second Derivative Test:

• If \( \frac{d^2y}{dx^2} > 0 \): Minimum point

• If \( \frac{d^2y}{dx^2} < 0 \): Maximum point

• If \( \frac{d^2y}{dx^2} = 0 \): Point of inflection (check further)

Standard Integrals

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

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

\[ \int e^x \, dx = e^x + C \]

\[ \int e^{kx} \, dx = \frac{1}{k}e^{kx} + C \]

\[ \int \sin x \, dx = -\cos x + C \]

\[ \int \cos x \, dx = \sin x + C \]

\[ \int \sec^2 x \, dx = \tan x + C \]

\[ \int \sin kx \, dx = -\frac{1}{k}\cos kx + C \]

\[ \int \cos kx \, dx = \frac{1}{k}\sin kx + C \]

Integration Techniques

Integration by Parts:

\[ \int u \frac{dv}{dx} \, dx = uv - \int v \frac{du}{dx} \, dx \]

Integration by Substitution:

\[ \int f(g(x))g'(x) \, dx = \int f(u) \, du \quad \text{where } u = g(x) \]

Definite Integration:

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

Area and Volume

Area under curve:

\[ A = \int_a^b y \, dx \]

Volume of revolution (about x-axis):

\[ V = \pi \int_a^b y^2 \, dx \]

Volume of revolution (about y-axis):

\[ V = \pi \int_c^d x^2 \, dy \]

Trigonometric Identities

Pythagorean Identities:

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

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

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

Reciprocal Identities:

\[ \sec \theta = \frac{1}{\cos \theta}, \quad \csc \theta = \frac{1}{\sin \theta}, \quad \cot \theta = \frac{1}{\tan \theta} \]

Compound Angle Formulas

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

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

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

Double Angle Formulas

\[ \sin 2A = 2\sin A \cos A \]

\[ \cos 2A = \cos^2 A - \sin^2 A = 2\cos^2 A - 1 = 1 - 2\sin^2 A \]

\[ \tan 2A = \frac{2\tan A}{1 - \tan^2 A} \]

Sine and Cosine Rules

Sine Rule:

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

Cosine Rule:

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

\[ \cos A = \frac{b^2 + c^2 - a^2}{2bc} \]

Area of Triangle:

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

R-Formula (Wave Function)

\[ a\cos \theta + b\sin \theta = R\cos(\theta - \alpha) \]

Where \( R = \sqrt{a^2 + b^2} \) and \( \tan \alpha = \frac{b}{a} \)

\[ a\cos \theta + b\sin \theta = R\sin(\theta + \alpha) \]

Where \( R = \sqrt{a^2 + b^2} \) and \( \tan \alpha = \frac{a}{b} \)

Straight Lines

Gradient: \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

Equation forms:

• \( y = mx + c \) (gradient-intercept form)

• \( y - y_1 = m(x - x_1) \) (point-slope form)

• \( ax + by + c = 0 \) (general form)

Distance between points:

\[ d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \]

Midpoint:

\[ M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right) \]

Perpendicular lines: \( m_1 \times m_2 = -1 \)

Parallel lines: \( m_1 = m_2 \)

Circles

Standard form: \( (x-a)^2 + (y-b)^2 = r^2 \)

Centre: \( (a, b) \), Radius: \( r \)

General form: \( x^2 + y^2 + 2gx + 2fy + c = 0 \)

Centre: \( (-g, -f) \), Radius: \( \sqrt{g^2 + f^2 - c} \)

Vector Operations

Magnitude: \( |\mathbf{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2} \)

Unit vector: \( \hat{\mathbf{a}} = \frac{\mathbf{a}}{|\mathbf{a}|} \)

Scalar (Dot) Product:

\[ \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}|\cos \theta \]

\[ \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \]

Vector (Cross) Product:

\[ \mathbf{a} \times \mathbf{b} = |\mathbf{a}||\mathbf{b}|\sin \theta \, \hat{\mathbf{n}} \]

Vector Equations

Line equation:

\[ \mathbf{r} = \mathbf{a} + \lambda\mathbf{b} \]

Where \( \mathbf{a} \) is a position vector and \( \mathbf{b} \) is the direction vector

Plane equation:

\[ \mathbf{r} \cdot \mathbf{n} = d \]

Where \( \mathbf{n} \) is the normal vector

Measures of Central Tendency

Mean: \( \bar{x} = \frac{\sum x}{n} = \frac{\sum fx}{\sum f} \)

Median: Middle value when data is ordered

For grouped data: \( L + \frac{\frac{n}{2} - F}{f} \times w \)

Mode: Most frequent value

Measures of Spread

Range: \( \text{Maximum} - \text{Minimum} \)

Interquartile Range (IQR): \( Q_3 - Q_1 \)

Variance:

\[ \sigma^2 = \frac{\sum(x - \bar{x})^2}{n} = \frac{\sum x^2}{n} - \bar{x}^2 \]

Standard Deviation:

\[ \sigma = \sqrt{\frac{\sum(x - \bar{x})^2}{n}} = \sqrt{\frac{\sum x^2}{n} - \bar{x}^2} \]

Sample variance: \( s^2 = \frac{\sum(x - \bar{x})^2}{n-1} \)

Probability Rules

\[ 0 \leq P(A) \leq 1 \]

\[ P(A') = 1 - P(A) \]

Addition Rule:

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

For mutually exclusive events: \( P(A \cup B) = P(A) + P(B) \)

Multiplication Rule:

\[ P(A \cap B) = P(A) \times P(B|A) \]

For independent events: \( P(A \cap B) = P(A) \times P(B) \)

Conditional Probability:

\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]

Binomial Distribution \( B(n, p) \)

\[ P(X = r) = \binom{n}{r} p^r (1-p)^{n-r} \]

Mean: \( E(X) = np \)

Variance: \( \text{Var}(X) = np(1-p) = npq \)

Normal Distribution \( N(\mu, \sigma^2) \)

Standardization:

\[ Z = \frac{X - \mu}{\sigma} \]

Where \( Z \sim N(0, 1) \)

Properties:

• \( E(X) = \mu \)

• \( \text{Var}(X) = \sigma^2 \)

Poisson Distribution \( \text{Po}(\lambda) \)

\[ P(X = r) = \frac{e^{-\lambda} \lambda^r}{r!} \]

Mean: \( E(X) = \lambda \)

Variance: \( \text{Var}(X) = \lambda \)

Test Statistics

Z-test (Normal distribution):

\[ Z = \frac{\bar{x} - \mu}{\sigma/\sqrt{n}} \]

T-test (Small samples):

\[ t = \frac{\bar{x} - \mu}{s/\sqrt{n}} \]

Degrees of freedom: \( \nu = n - 1 \)

Confidence Intervals

For population mean (\( \sigma \) known):

\[ \bar{x} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}} \]

For population mean (\( \sigma \) unknown):

\[ \bar{x} \pm t_{\alpha/2} \frac{s}{\sqrt{n}} \]

Correlation Coefficient

Product Moment Correlation Coefficient (PMCC):

\[ r = \frac{S_{xy}}{\sqrt{S_{xx} S_{yy}}} = \frac{\sum(x-\bar{x})(y-\bar{y})}{\sqrt{\sum(x-\bar{x})^2 \sum(y-\bar{y})^2}} \]

Where:

\[ S_{xx} = \sum(x-\bar{x})^2 = \sum x^2 - \frac{(\sum x)^2}{n} \]

\[ S_{yy} = \sum(y-\bar{y})^2 = \sum y^2 - \frac{(\sum y)^2}{n} \]

\[ S_{xy} = \sum(x-\bar{x})(y-\bar{y}) = \sum xy - \frac{\sum x \sum y}{n} \]

Linear Regression

Regression line of y on x:

\[ y = a + bx \]

Where:

\[ b = \frac{S_{xy}}{S_{xx}} \]

\[ a = \bar{y} - b\bar{x} \]

SUVAT Equations (Constant Acceleration)

\[ v = u + at \]

\[ s = ut + \frac{1}{2}at^2 \]

\[ v^2 = u^2 + 2as \]

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

\[ s = vt - \frac{1}{2}at^2 \]

Where: \( s \) = displacement, \( u \) = initial velocity, \( v \) = final velocity, \( a \) = acceleration, \( t \) = time

Motion Graphs

Velocity: \( v = \frac{ds}{dt} \)

Acceleration: \( a = \frac{dv}{dt} = \frac{d^2s}{dt^2} \)

Displacement from velocity-time graph:

\[ s = \int v \, dt \]

Projectile Motion

Horizontal component: \( u_x = u\cos\theta \)

Vertical component: \( u_y = u\sin\theta \)

Time of flight: \( T = \frac{2u\sin\theta}{g} \)

Maximum height: \( H = \frac{u^2\sin^2\theta}{2g} \)

Range: \( R = \frac{u^2\sin 2\theta}{g} \)

Newton's Laws

First Law: An object remains at rest or in uniform motion unless acted upon by a force

Second Law:

\[ F = ma \]

\[ F = \frac{d(mv)}{dt} \]

Third Law: For every action, there is an equal and opposite reaction

Weight and Friction

Weight: \( W = mg \)

Friction: \( F \leq \mu R \)

Where \( \mu \) = coefficient of friction, \( R \) = normal reaction

Limiting friction: \( F_{\text{max}} = \mu R \)

Work and Energy

Work done: \( W = Fs\cos\theta \)

Kinetic Energy:

\[ KE = \frac{1}{2}mv^2 \]

Potential Energy:

\[ PE = mgh \]

Work-Energy Theorem:

\[ W = \Delta KE \]

Conservation of Energy:

\[ KE + PE = \text{constant} \]

Power

\[ P = \frac{W}{t} = \frac{E}{t} \]

\[ P = Fv \]

Where \( P \) = power, \( W \) = work done, \( E \) = energy, \( F \) = force, \( v \) = velocity

Momentum

Momentum: \( p = mv \)

Impulse:

\[ I = Ft = \Delta p = m(v - u) \]

Conservation of Momentum:

\[ m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2 \]

Collisions

Coefficient of Restitution:

\[ e = \frac{\text{relative speed after collision}}{\text{relative speed before collision}} = \frac{v_2 - v_1}{u_1 - u_2} \]

• \( e = 1 \): Perfectly elastic collision

• \( 0 < e < 1 \): Inelastic collision

• \( e = 0 \): Perfectly inelastic collision

Moments

Moment of a force:

\[ M = Fd \]

Where \( d \) = perpendicular distance from pivot

Principle of Moments:

For equilibrium: \( \sum \text{clockwise moments} = \sum \text{anticlockwise moments} \)

Conditions for Equilibrium:

• \( \sum F_x = 0 \)

• \( \sum F_y = 0 \)

• \( \sum M = 0 \)

Do I get a formula sheet in A Level Maths exams?

Yes, all major exam boards (Edexcel, AQA, OCR, Cambridge, SEAB) provide official formula booklets during A Level Maths exams. However, not all formulas are included, so you must memorize certain key formulas.

What's the difference between exam board formula booklets?

Edexcel provides the most comprehensive booklet covering Pure, Statistics, and Mechanics. AQA has smaller booklets with fewer formulas. OCR organizes by module. Content is similar across boards since the 2017 syllabus reform.

Which formulas are NOT in the booklet?

Common exclusions: basic geometric formulas (circle equations, straight line equations), fundamental trigonometric identities, GCSE-level formulas, basic differentiation rules, and simple algebra formulas. Always check your specific exam board's booklet.

Can I write on my formula booklet during the exam?

Yes, you can annotate your formula booklet during the exam. Many students find it helpful to mark frequently used formulas or add notes.

Is the Further Maths formula booklet different?

Yes, Further Mathematics has an extended formula booklet that includes additional topics like complex numbers, matrices, differential equations, and advanced mechanics. It's larger than the standard A Level Maths booklet.