Edexcel A Level Maths Formulas

Includes every Edexcel A Level Maths formula, generated by AI.

Quadratic formula (roots of ax^2 + bx + c = 0)

x = (-b ± √(b^2 - 4ac)) / (2a)

Discriminant of a quadratic (ax^2 + bx + c)

Δ = b^2 - 4ac

Laws of indices: multiplication

a^x × a^y ≡ a^(x + y)

Laws of indices: division

a^x ÷ a^y ≡ a^(x - y)

Laws of indices: power of a power

(a^x)^y ≡ a^(xy)

Laws of indices: fractional powers

a^(m/n) ≡ ⁿ√(a^m)

Definition of a logarithm

log_a(x) = n ⇔ a^n = x (for a > 0, x > 0)

Laws of logarithms: product rule

log_a(x) + log_a(y) ≡ log_a(xy)

Laws of logarithms: quotient rule

log_a(x) - log_a(y) ≡ log_a(x / y)

Laws of logarithms: power rule

k log_a(x) ≡ log_a(x^k)

Change of base formula for logarithms

log_a(x) = log_b(x) / log_b(a)

Exponential and natural log inverse identity

e^(x ln a) = a^x

Equation of a straight line (point-gradient form)

y - y_1 = m(x - x_1)

Gradient condition for perpendicular lines

m_1 × m_2 = -1

Distance between two points in 2D

d = √((x_1 - x_2)^2 + (y_1 - y_2)^2)

Distance between two points in 3D

d = √((x_1 - x_2)^2 + (y_1 - y_2)^2 + (z_1 - z_2)^2)

Equation of a circle with centre (a, b) and radius r

(x - a)^2 + (y - b)^2 = r^2

Arithmetic progression: nth term

u_n = a + (n - 1)d

Arithmetic series: sum to n terms

S_n = (1/2)n(a + l) = (1/2)n[2a + (n - 1)d]

Geometric progression: nth term

u_n = ar^(n - 1)

Geometric series: sum to n terms

S_n = a(1 - r^n) / (1 - r)

Geometric series: sum to infinity (|r| < 1)

S_∞ = a / (1 - r)

Binomial coefficient (nCr)

nCr = n! / (r!(n - r)!)

Binomial expansion (positive integer n)

(a + b)^n = a^n + nC1 a^(n-1)b + nC2 a^(n-2)b^2 + … + nCr a^(n-r)b^r + … + b^n

Binomial expansion (rational n, |x| < 1)

(1 + x)^n = 1 + nx + [n(n - 1) / 2!]x^2 + … + [n(n - 1)…(n - r + 1) / r!]x^r + …

Sine rule

a / sin A = b / sin B = c / sin C

Cosine rule

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

Area of a triangle (trigonometric)

Area = (1/2)ab sin C

Fundamental trigonometric identity

cos^2(A) + sin^2(A) ≡ 1

Tangent quotient identity

tan θ ≡ sin θ / cos θ

Secant Pythagorean identity

sec^2(A) ≡ 1 + tan^2(A)

Cosecant Pythagorean identity

cosec^2(A) ≡ 1 + cot^2(A)

Double angle identity: sin(2A)

sin(2A) ≡ 2 sin A cos A

Double angle identity: cos(2A)

cos(2A) ≡ cos^2(A) - sin^2(A) ≡ 2cos^2(A) - 1 ≡ 1 - 2sin^2(A)

Double angle identity: tan(2A)

tan(2A) ≡ 2 tan A / (1 - tan^2(A))

Compound angle identity: sin(A ± B)

sin(A ± B) = sin A cos B ± cos A sin B

Compound angle identity: cos(A ± B)

cos(A ± B) = cos A cos B ∓ sin A sin B

Compound angle identity: tan(A ± B)

tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B)

Factor formula: sin A + sin B

sin A + sin B = 2 sin((A + B)/2) cos((A - B)/2)

Factor formula: sin A - sin B

sin A - sin B = 2 cos((A + B)/2) sin((A - B)/2)

Factor formula: cos A + cos B

cos A + cos B = 2 cos((A + B)/2) cos((A - B)/2)

Factor formula: cos A - cos B

cos A - cos B = -2 sin((A + B)/2) sin((A - B)/2)

Small angle approximation: sin θ (radians)

sin θ ≈ θ

Small angle approximation: cos θ (radians)

cos θ ≈ 1 - (θ^2 / 2)

Small angle approximation: tan θ (radians)

tan θ ≈ θ

Circumference of a circle

C = 2πr = πd

Area of a circle

A = πr^2

Surface area of a sphere

Surface area = 4πr^2

Curved surface area of a cone

Area = πr × slant height

Area of a trapezium

Area = (1/2)(a + b)h

Volume of a prism

Volume = area of cross-section × length

Arc length of a sector (θ in radians)

s = rθ

Area of a sector (θ in radians)

A = (1/2)r^2 θ

Differentiation from first principles

f'(x) = lim(h→0) [f(x + h) - f(x)] / h

Derivative of x^n

d/dx (x^n) = nx^(n - 1)

Derivative of sin(kx)

d/dx (sin kx) = k cos kx

Derivative of cos(kx)

d/dx (cos kx) = -k sin kx

Derivative of tan(kx)

d/dx (tan kx) = k sec^2(kx)

Derivative of sec(kx)

d/dx (sec kx) = k sec kx tan kx

Derivative of cot(kx)

d/dx (cot kx) = -k cosec^2(kx)

Derivative of cosec(kx)

d/dx (cosec kx) = -k cosec kx cot kx

Derivative of e^(kx)

d/dx (e^(kx)) = k e^(kx)

Derivative of a^(kx)

d/dx (a^(kx)) = k a^(kx) ln a

Derivative of ln(x)

d/dx (ln x) = 1 / x

Product rule for differentiation

d/dx [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)

Quotient rule for differentiation

d/dx [f(x) / g(x)] = [f'(x)g(x) - f(x)g'(x)] / [g(x)]^2

Chain rule for differentiation

d/dx [f(g(x))] = f'(g(x))g'(x)

Parametric differentiation

dy/dx = (dy/dt) / (dx/dt)

Integral of x^n (n ≠ -1)

∫ x^n dx = [x^(n + 1) / (n + 1)] + c

Integral of cos(kx)

∫ cos kx dx = (1/k) sin kx + c

Integral of sin(kx)

∫ sin kx dx = -(1/k) cos kx + c

Integral of sec^2(kx)

∫ sec^2(kx) dx = (1/k) tan kx + c

Integral of tan(kx)

∫ tan kx dx = (1/k) ln|sec kx| + c

Integral of cot(kx)

∫ cot kx dx = (1/k) ln|sin kx| + c

Integral of cosec(kx)

∫ cosec kx dx = -(1/k) ln|cosec kx + cot kx| + c = (1/k) ln|tan(kx/2)| + c

Integral of sec(kx)

∫ sec kx dx = (1/k) ln|sec kx + tan kx| + c = (1/k) ln|tan((kx/2) + (π/4))| + c

Integral of e^(kx)

∫ e^(kx) dx = (1/k) e^(kx) + c

Integral of 1/x (x ≠ 0)

∫ (1/x) dx = ln|x| + c

Reverse chain rule: ∫ f'(g(x))g'(x) dx

∫ f'(g(x))g'(x) dx = f(g(x)) + c

Reverse chain rule: ∫ f'(x)/f(x) dx

∫ [f'(x) / f(x)] dx = ln|f(x)| + c

Integration by parts

∫ u (dv/dx) dx = uv - ∫ v (du/dx) dx

Area under a curve (y ≥ 0)

Area = ∫_a^b y dx

Trapezium rule for numerical integration

∫a^b y dx ≈ (1/2)h[(y_0 + y_n) + 2(y_1 + y_2 + … + y(n-1))], where h = (b - a)/n

Newton-Raphson iteration formula

x_(n+1) = x_n - [f(x_n) / f'(x_n)]

Magnitude of a 3D vector xi + yj + zk

|xi + yj + zk| = √(x^2 + y^2 + z^2)

Mean of a set of data

x̄ = (Σx / n) = (Σfx / Σf)

Interquartile range (IQR)

IQR = Q_3 - Q_1

Summary statistic S_xx

S_xx = Σ(x_i - x̄)^2 = Σx_i^2 - [(Σx_i)^2 / n]

Population standard deviation

σ = √(Variance) = √(S_xx / n) = √[(Σx^2 / n) - x̄^2]

Probability of complementary event

P(A') = 1 - P(A)

Addition rule of probability

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

Multiplication rule of probability

P(A ∩ B) = P(A)P(B | A)

Conditional probability formula

P(A | B) = P(A ∩ B) / P(B)

Independent events conditions

P(B | A) = P(B), P(A | B) = P(A), and P(A ∩ B) = P(A)P(B)

Binomial distribution probability mass function

P(X = x) = nCx p^x (1 - p)^(n - x)

Binomial distribution mean and variance

Mean = np, Variance = np(1 - p)

Standard Normal variable transformation

Z = (X - μ) / σ ~ N(0, 1)

Sample mean Normal distribution transformation

Z = (X̄ - μ) / (σ / √n) ~ N(0, 1)

SUVAT equation 1 (no s)

v = u + at

SUVAT equation 2 (no v)

s = ut + (1/2)at^2