Equations

Equation index

Every result used across the notes, grouped by unit and linked back to the lesson it came from. Each is marked according to whether the Edexcel formulae booklet prints it or you are expected to carry it in your head.

The 45 the booklet does not give you

These are the results you have to know outright. If revision time is short, this is the list to start from, because the rest can be looked up in the exam.

Every result, by unit

Jump to: Algebra and functions · Coordinate geometry · Sequences and series · Trigonometry · Exponentials and logarithms · Differentiation · Integration · Numerical methods · Vectors · Statistics · Mechanics · Further proof · Complex numbers · Further algebra and series · Further vectors · Further calculus · Hyperbolic functions · Polar coordinates · Differential equations · Further Pure 1 · Further Statistics 1 · Further Pure 2 · Further Statistics 2 · Further Mechanics 1 · Further Mechanics 2 · Decision Mathematics 1 · Decision Mathematics 2

Algebra and functions7 results

Indices and surds

a^x × a^y = a^x+yNOT IN THE BOOKLET — LEARN IT

a^x ÷ a^y = a^x-yNOT IN THE BOOKLET — LEARN IT

(a^x)^y = a^xyNOT IN THE BOOKLET — LEARN IT

√xy = √x√yNOT IN THE BOOKLET — LEARN IT

Quadratic functions

ax² + bx + c = a(x + b2a)² + (c − b²4a)

x = −b ± √b² − 4ac2aNOT IN THE BOOKLET — LEARN IT

Polynomials and the factor theorem

f(a) = 0 ⇔ (x − a) is a factor of f(x)

Coordinate geometry3 results

Straight lines

y − y₁ = m(x − x₁)NOT IN THE BOOKLET — LEARN IT

m₁ m₂ = −1NOT IN THE BOOKLET — LEARN IT

Circles

(x − a)² + (y − b)² = r²

Sequences and series8 results

The binomial expansion

ⁿC_r = n!r!(n − r)!IN THE FORMULAE BOOKLET

(a + b)ⁿ = aⁿ + ⁿC₁aⁿ⁻¹b + … + bⁿIN THE FORMULAE BOOKLET

The general binomial expansion

(1 + x)ⁿ = 1 + nx + n(n − 1)2!x² + …IN THE FORMULAE BOOKLET

Arithmetic series

u_n = a + (n − 1)dNOT IN THE BOOKLET — LEARN IT

S_n = n2(2a + (n − 1)d) = n2(a + l)IN THE FORMULAE BOOKLET

Geometric series

u_n = arⁿ⁻¹NOT IN THE BOOKLET — LEARN IT

S_n = a(1 − rⁿ)1 − rIN THE FORMULAE BOOKLET

S_∞ = a1 − rIN THE FORMULAE BOOKLET

Trigonometry12 results

Triangles and the sine and cosine rules

asin A = bsin B = csin CNOT IN THE BOOKLET — LEARN IT

a² = b² + c² − 2bc cos ANOT IN THE BOOKLET — LEARN IT

Area = 12ab sin CNOT IN THE BOOKLET — LEARN IT

Trigonometric graphs and equations

tan θ = sin θcos θNOT IN THE BOOKLET — LEARN IT

sin ² θ + cos ² θ = 1NOT IN THE BOOKLET — LEARN IT

Radians, arcs and small angles

s = rθNOT IN THE BOOKLET — LEARN IT

A = 12r²θNOT IN THE BOOKLET — LEARN IT

sin θ ≈ θ, cos θ ≈ 1 − θ²2, tan θ ≈ θNOT IN THE BOOKLET — LEARN IT

Reciprocal and inverse trigonometric functions

sec² θ = 1 + tan² θNOT IN THE BOOKLET — LEARN IT

cosec² θ = 1 + cot² θNOT IN THE BOOKLET — LEARN IT

Compound angles and the harmonic form

sin (A ± B) = sin A cos B ± cos A sin BIN THE FORMULAE BOOKLET

cos (A ± B) = cos A cos B ∓ sin A sin BIN THE FORMULAE BOOKLET

Exponentials and logarithms5 results

Exponential functions and e

gradient of e^kx = ke^kxNOT IN THE BOOKLET — LEARN IT

Logarithms and their laws

x = aⁿ ⇔ n = log_a xNOT IN THE BOOKLET — LEARN IT

log_a x + log_a y = log_a xyNOT IN THE BOOKLET — LEARN IT

log_a x − log_a y = log_a xyNOT IN THE BOOKLET — LEARN IT

k log_a x = log_a x^kNOT IN THE BOOKLET — LEARN IT

Differentiation7 results

The derivative from first principles

f'(x) = lim_{h → 0} f(x + h) − f(x)h

Differentiating powers of x

y = xⁿ ⇒ dydx = nxⁿ⁻¹NOT IN THE BOOKLET — LEARN IT

Differentiating trig, exponentials and logs

sin kx → kcos kx, cos kx → −ksin kxNOT IN THE BOOKLET — LEARN IT

e^kx → ke^kx, ln x → 1xNOT IN THE BOOKLET — LEARN IT

The product, quotient and chain rules

dydx = dydu × dudxNOT IN THE BOOKLET — LEARN IT

(uv)' = u'v − uv'v²IN THE FORMULAE BOOKLET

Implicit and parametric differentiation

dydx = 1dx/dy = 1cos y

Integration11 results

Integration as antidifferentiation

∫ xⁿ dx = xⁿ⁺¹n + 1 + c (n ≠ −1)NOT IN THE BOOKLET — LEARN IT

Integrating standard functions

∫ cos kx dx = 1ksin kx + cNOT IN THE BOOKLET — LEARN IT

∫ sin kx dx = −1kcos kx + cNOT IN THE BOOKLET — LEARN IT

∫ e^kx dx = 1ke^kx + cNOT IN THE BOOKLET — LEARN IT

∫ 1x dx = ln|x| + cNOT IN THE BOOKLET — LEARN IT

Integration by substitution and by parts

∫ f'(x)f(x) dx = ln|f(x)| + cNOT IN THE BOOKLET — LEARN IT

∫ u dvdx dx = uv − ∫ v dudx dxIN THE FORMULAE BOOKLET

Integrating rational functions

∫ 23x + 5 dx = 23ln|3x + 5| + c

Areas, parametric curves and the limit of a sum

area = ∫ y dxdt dt

lim_{δx → 0} Σ f(x) δx = ∫_a^b f(x) dx

Solving differential equations

∫ 1g(y) dy = ∫ f(x) dx

Numerical methods3 results

Locating roots and iteration

x_n+1 = (x_n + 1)^1/3

Newton-Raphson and the trapezium rule

x_n+1 = x_n − f(x_n)f'(x_n)IN THE FORMULAE BOOKLET

T = h2(y₀ + 2(y₁ + … + y_n−1) + y_n)IN THE FORMULAE BOOKLET

Vectors2 results

Vectors in two dimensions

AB = b − aNOT IN THE BOOKLET — LEARN IT

Vectors in three dimensions

|xi + yj + zk| = √x² + y² + z²NOT IN THE BOOKLET — LEARN IT

Statistics11 results

Sampling and the large data set

stratum sizepopulation size × sample size

Measures of location and spread

mean: Σ xnNOT IN THE BOOKLET — LEARN IT

σ² = Σ x²n - (Σ xn)²IN THE FORMULAE BOOKLET

mean(x) = a + b mean(y),     σ_x = bσ_yNOT IN THE BOOKLET — LEARN IT

Representing and interpreting data

frequency density = frequencyclass widthNOT IN THE BOOKLET — LEARN IT

outlier: beyond Q₁ - 1.5 × IQR or Q₃ + 1.5 × IQRNOT IN THE BOOKLET — LEARN IT

Probability and Venn diagrams

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)IN THE FORMULAE BOOKLET

Conditional probability

P(A|B) = P(A ∩ B)P(B)IN THE FORMULAE BOOKLET

The binomial distribution

P(X = x) = ⁿC_x p^x (1 - p)^{n - x}IN THE FORMULAE BOOKLET

The normal distribution

Z = X - μσNOT IN THE BOOKLET — LEARN IT

Hypothesis testing: correlation and the normal

mean of n observations: N(μ, σ²/n)IN THE FORMULAE BOOKLET

Mechanics6 results

Kinematics with constant acceleration

v = u + at,     s = (u + v)2tIN THE FORMULAE BOOKLET

s = ut + 12at²,     v² = u² + 2asIN THE FORMULAE BOOKLET

Kinematics with variable acceleration

v = dxdt,     a = dvdtNOT IN THE BOOKLET — LEARN IT

Forces and Newton's laws

F = maNOT IN THE BOOKLET — LEARN IT

Friction and inclined planes

F ≤ μRNOT IN THE BOOKLET — LEARN IT

Moments

moment = force × perpendicular distanceNOT IN THE BOOKLET — LEARN IT

Further proof1 result

Proof by induction: sums and series

true for n = 1, and true for k + 1 whenever true for k

Complex numbers2 results

Modulus, argument and loci

z = r(cos θ + i sin θ)

De Moivre's theorem and trigonometric identities

(cos θ + i sin θ)ⁿ = cos nθ + i sin nθ

Further algebra and series4 results

Roots of polynomials

α + β + γ = -ba, αβ + αγ + βγ = ca, αβγ = -da

Summing series and the method of differences

Σ r = n(n+1)2, Σ r² = n(n+1)(2n+1)6, Σ r³ = n²(n+1)²4

Maclaurin series

f(x) = f(0) + f'(0)x + f''(0)2!x² + …

ln(1 + x) = x - x²2 + x³3 - x⁴4 + … (-1 < x ≤ 1)

Further vectors3 results

Lines and planes in three dimensions

r = a + λb

r · n = a · n = d

Intersections, angles and distances

distance = |n₁x₀ + n₂y₀ + n₃z₀ - d||n|

Further calculus4 results

Volumes of revolution

V = π ∫_a^b y² dx

Mean values and improper integrals

mean value = 1b - a ∫_a^b f(x) dx

Calculus with inverse trigonometric functions

ddx(arcsin x) = 1√1 - x²

ddx(arctan x) = 11 + x²

Hyperbolic functions2 results

Hyperbolic functions and identities

cosh x = e^x + e^-x2, sinh x = e^x - e^-x2

Calculus with hyperbolic functions

∫ 1√x² + a² dx = arsinhxa + c

Polar coordinates2 results

Polar curves

x = r cos θ, y = r sin θ, r² = x² + y²

Areas with polar coordinates

A = 12 ∫_α^β r² dθ

Differential equations2 results

First order equations and integrating factors

IF = e^{∫ P dx}

Second order equations

am² + bm + c = 0

Further Pure 17 results

The t-formulae

sin θ = 2t1 + t², cos θ = 1 - t²1 + t², tan θ = 2t1 - t²

Taylor series

f(x) = f(a) + f'(a)(x - a) + f''(a)2!(x - a)² + …

Leibnitz's theorem and the Weierstrass substitution

(fg)⁽ⁿ⁾ = Σ ⁿC_r f^(r) g^(n-r)

Conic sections

x²a² + y²b² = 1, x = a cos t, y = b sin t

The vector product and the scalar triple product

a × b = (a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁)

Numerical methods for differential equations

dydx ≈ y_n+1 - y_nh or y_n+1 - y_n-12h

∫_a^b y dx ≈ h3(y₀ + 4y₁ + 2y₂ + … + 4y_n-1 + y_n)

Further Statistics 111 results

Discrete random variables and expectation

E(X) = Σ x P(X = x)

Var(X) = Σ x² P(X = x) - μ²

The Poisson distribution

P(X = x) = e^-λ λ^xx!, E(X) = Var(X) = λ

Geometric and negative binomial distributions

P(X = x) = p(1 - p)^x-1, μ = 1p, σ² = 1 - pp²

P(X = x) = ^x-1C_r-1 p^r(1 - p)^x-r

μ = rp, σ² = r(1 - p)p²

The Central Limit Theorem

sample mean ≈ N(μ, σ²n)

Goodness-of-fit tests

χ² = Σ (O_i - E_i)²E_i

Contingency tables

E_ij = row total × column totalgrand total

Probability generating functions

G(t) = E(t^X) = Σ t^x P(X = x)

Var(X) = G''(1) + G'(1) - [G'(1)]²

Further Pure 25 results

Reduction formulae

n I_n = (n - 1) I_n-2, I_n = ∫₀^π/2 sin ⁿ x dx

Arc length and surface area

s = ∫ √1 + (dydx)² dx

Eigenvalues and eigenvectors

det(M - λI) = 0

Diagonalisation and the Cayley-Hamilton theorem

P^-1MP = D

Modular arithmetic and Fermat's little theorem

a^p-1 ≡ 1 (mod p)

Further Statistics 217 results

Least squares regression and residuals

b = S_xyS_xx,     a = mean of y - b × mean of x

RSS = S_yy - (S_xy)²S_xx

Continuous random variables: density and distribution functions

P(a < X ≤ b) = ∫_a^b f(x) dx

f(x) = dF(x)dx

Mean, variance and skewness of continuous variables

E(X) = ∫ x f(x) dx,      Var(X) = E(X²) - [E(X)]²

The continuous uniform distribution

F(x) = x - ab - a for a ≤ x ≤ b

E(X) = a + b2,      Var(X) = (b - a)²12

Correlation coefficients: product moment and Spearman

r = S_xy√S_xx S_yy

r_s = 1 - 6 Σ d²n(n² - 1)

Combinations of normal random variables

aX ± bY is N(aμ_x ± bμ_y, a²σ_x² + b²σ_y²)

Estimators, standard error and confidence intervals

s² = 1n - 1 Σ (x_i - sample mean)²

sample mean ± z × σ√n,     z = 1.96 at 95 per cent

Comparing two normal means

(difference of means) - (μ_x - μ_y)√σ_x²n_x + σ_y²n_y is N(0, 1)

Testing variances: chi-squared and the F-distribution

(n - 1)S²σ² is χ² on n - 1 degrees of freedom

S₁²S₂² is F on n₁ - 1 and n₂ - 1 degrees of freedom

Confidence intervals and tests with the t-distribution

sample mean - μS√n is t on n - 1 degrees of freedom

s² = (n₁ - 1)s₁² + (n₂ - 1)s₂²n₁ + n₂ - 2

Further Mechanics 18 results

Momentum and impulse

I = Ft = mv - mu

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

Impulse and momentum as vectors

I = mv - mu

Work, energy and power

KE = 12mv²,      GPE = mgh

P = Fv

Hooke's law and elastic strings

T = λ xl

Elastic potential energy

EPE = λ x²2l

Direct impact and Newton's law of restitution

e = speed of separationspeed of approach

Further Mechanics 211 results

Angular speed and horizontal circular motion

a = rω² = v²r

tan θ = v²rg

Motion in a vertical circle

T - mgcos θ = mv²r

Centre of mass of a discrete distribution

mean x = Σ m_ix_iΣ m_i,      mean y = Σ m_iy_iΣ m_i

Centres of mass by integration

mean x = ∫ xy dx∫ y dx,      mean y = ∫ 12y² dx∫ y dx

mean x = ∫ x y² dx∫ y² dx

Newton's laws with a variable force

F(x) = mvdvdx,     F(t) = mdvdt

Simple harmonic motion

d²xdt² = -ω²x

v² = ω²(a² - x²)

Oscillations on strings and springs

md²xdt² = -λ xl,     ω² = λml

Further kinematics: acceleration as a function of x, t or v

dvdt = f(t) or f(v),     vdvdx = f(v) or f(x)

Decision Mathematics 11 result

Linear programming: formulation and graphical solution

3x + 2y ≤ 20 becomes 3x + 2y + s₁ = 20

Decision Mathematics 21 result

The stepping-stone method

improvement index = cost - R - K

156 results in total, 45 of them off the booklet. See also the key ideas, command words and the revision checklist.