ECZ Mathematics formula sheet

Difference of two squares

a² − b² = (a − b)(a + b)

Use it when: Two perfect squares with a minus between them.

By far the most-asked factorisation in Paper 1.

Perfect square

(a + b)² = a² + 2ab + b²

Use it when: Expanding a bracket squared, or spotting one in reverse.

Perfect square (minus)

(a − b)² = a² − 2ab + b²

Use it when: As above, with a subtraction.

Multiplying powers

aᵐ × aⁿ = aᵐ⁺ⁿ

Use it when: The bases are the same and you are multiplying — add the powers.

Dividing powers

aᵐ ÷ aⁿ = aᵐ⁻ⁿ

Use it when: Same base, dividing — subtract the powers.

Power of a power

(aᵐ)ⁿ = aᵐⁿ

Use it when: A power is raised to another power — multiply them.

Zero index

a⁰ = 1

Use it when: Anything (except 0) to the power zero.

A favourite trick in Paper 1 — spot it and the question collapses.

Negative index

a⁻ⁿ = 1aⁿ

Use it when: You need to clear a negative power — flip it into a fraction.

Fractional index

a^m/n = ⁿ√(aᵐ)

Use it when: The power is a fraction — the denominator is the root.

Determinant of a 2×2

det = ad − bc

Use it when: For the matrix (a b; c d) — needed before any inverse.

Inverse of a 2×2

A⁻¹ = 1ad − bc× (d −b; −c a)

Use it when: Solving matrix equations.

Swap a and d, change the signs of b and c, then divide by the determinant.

Singular matrix

ad − bc = 0

Use it when: A matrix with no inverse — the determinant is zero.

Union of two sets

n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

Use it when: Counting members of two overlapping sets — the overlap was counted twice.

Complement

n(A′) = n(ε) − n(A)

Use it when: You need everything outside a set.

Ratio of areas

area ratio = k²

Use it when: Two similar shapes with length scale factor k.

Ratio of volumes

volume ratio = k³

Use it when: Two similar solids with length scale factor k.

Lengths k, areas k², volumes k³ — the step most often forgotten.

Simple interest

I = P × R × T100

Use it when: Interest is on the original amount only.

Compound interest

A = P(1 + R100)ⁿ

Use it when: Interest earns interest — n is the number of periods.

A is the total amount; interest alone is A − P.

Percentage profit

% profit = (profitcostprice) × 100

Use it when: Profit or loss as a percentage — always over COST price.

Discount

sale price = marked price − (discount% /100 × marked price)

Use it when: A price is reduced by a percentage.

Speed

speed = distancetime

Use it when: Gradient of a distance–time graph.

Acceleration

a = v − ut

Use it when: Gradient of a velocity–time graph.

Distance from a velocity–time graph

distance = area under the graph

Use it when: Split it into triangles and trapezia and add them.

Standard form

A × 10ⁿ, 1 ≤ A < 10

Use it when: Writing very large or very small numbers.

A must be at least 1 and less than 10 — that single condition decides most answers.

Percentage error

% error = (erroractualvalue) × 100

Use it when: Comparing a measurement with the true value.

Absolute error

error = measured value − actual value

Use it when: Take the size of the difference, ignoring sign.

Gradient

m = y₂ − y₁x₂ − x₁

Use it when: Steepness of a line through two points.

Equation of a line

y = mx + c

Use it when: m is the gradient, c is where it cuts the y-axis.

Midpoint

M = (x₁ + x₂2, y₁ + y₂2)

Use it when: The point exactly halfway between two points.

Distance between two points

d = √((x₂ − x₁)² + (y₂ − y₁)²)

Use it when: Length of the line joining two points — Pythagoras in disguise.

Perpendicular lines

m₁ × m₂ = −1

Use it when: Two lines meet at 90°.

Circle

Area = πr², Circumference = 2πr

Use it when: Any full circle.

Arc length

L = (θ/360) × 2πr

Use it when: Part of a circle's edge, θ is the angle at the centre.

Sector area

A = (θ/360) × πr²

Use it when: A slice of a circle.

Trapezium

Area = 12(a + b)h

Use it when: Four sides with one pair parallel — a and b are the parallel sides.

Cylinder

V = πr²h, curved surface = 2πrh

Use it when: A tin, a pipe, a tank.

Cone

V = 13πr²h, curved surface = πrl

Use it when: l is the slant height, not the vertical height.

Sphere

V = 43πr³, surface area = 4πr²

Use it when: A ball or a dome (half of it for a hemisphere).

Pyramid

V = 13× base area × height

Use it when: Any pyramid or cone-like solid with a flat base.

Probability of an event

P(A) = number of favourable outcomestotalnumber of outcomes

Use it when: All outcomes are equally likely.

Complement

P(A′) = 1 − P(A)

Use it when: 'At least one' questions — find the opposite and subtract.

Independent events (and)

P(A and B) = P(A) × P(B)

Use it when: One event does not affect the other — multiply along the branches.

Mutually exclusive (or)

P(A or B) = P(A) + P(B)

Use it when: The events cannot both happen — add the branches.

Without replacement

the second fraction has one fewer in the total

Use it when: An item is not put back — the denominator drops by 1.

This is where most marks are lost in tree-diagram questions.

Quadratic formula

x = (−b ± √(b² − 4ac))/(2a)

Use it when: ax² + bx + c = 0 will not factorise neatly.

Discriminant

b² − 4ac

Use it when: Deciding how many roots there are: positive → two, zero → one, negative → none.

AP — nth term

Tₙ = a + (n − 1)d

Use it when: The sequence goes up (or down) by a fixed amount d.

AP — sum of n terms

Sₙ = n2[2a + (n − 1)d]

Use it when: Adding up the first n terms of an AP.

GP — nth term

Tₙ = arⁿ⁻¹

Use it when: Each term is the previous one times a fixed ratio r.

GP — sum of n terms

Sₙ = arⁿ − 1r − 1

Use it when: Adding up the first n terms of a GP.

GP — sum to infinity

S∞ = a1 − r

Use it when: A GP that shrinks — only valid when −1 < r < 1.

Mean

x̄ = Σxn

Use it when: A plain list of values.

Mean from a frequency table

x̄ = Σfx/Σf

Use it when: Values come with frequencies — x is the midpoint for grouped data.

Standard deviation

σ = √(Σf(x − x̄)²/Σf)

Use it when: Measuring spread about the mean.

Standard deviation (working form)

σ = √(Σfx²/Σf − (Σfx/Σf)²)

Use it when: Usually faster in the exam — same answer, fewer steps.

Quartiles from a cumulative frequency curve

Q₁ at 14N, median at 12N, Q₃ at 34N

Use it when: Reading a cumulative frequency (ogive) graph.

Interquartile range = Q₃ − Q₁.

Pythagoras

a² + b² = c²

Use it when: Right-angled triangle, no angles needed — c is the hypotenuse.

SOHCAHTOA

sin θ = opphyp, cos θ = adjhyp, tan θ = oppadj

Use it when: Right-angled triangle with an angle involved.

Sine rule

asinA = bsinB = csinC

Use it when: Non-right triangle: two angles and a side, or two sides and a non-included angle.

Cosine rule

a² = b² + c² − 2bc cos A

Use it when: Non-right triangle: two sides and the angle between them, or all three sides.

Area of a triangle

Area = 12ab sin C

Use it when: Two sides and the angle between them — no height given.

Direct variation

y = kx

Use it when: y increases as x increases, in proportion.

Inverse variation

y = kx

Use it when: y decreases as x increases.

Joint variation

y = kxz

Use it when: y depends on two quantities together.

Partial variation

y = kx + c

Use it when: Part is fixed and part varies.

Distance along a great circle

d = (θ/360) × 2πR

Use it when: Along a meridian, or along the equator. R ≈ 6370 km.

Distance along a parallel of latitude

d = (θ/360) × 2πR cos α

Use it when: Travelling east–west at latitude α — the circle is smaller.

Distance in nautical miles

d = 60θ

Use it when: Along a great circle, θ in degrees.

Nautical miles along a parallel

d = 60θ cos α

Use it when: East–west at latitude α.

Longitude and time

15° of longitude = 1 hour

Use it when: Local time differences — east is ahead.

Translation

image = object + (a; b)

Use it when: Every point slides by the same column vector.

Enlargement

image distance = k × object distance from the centre

Use it when: k is the scale factor; negative k puts the image on the opposite side.

Area after enlargement

new area = k² × old area

Use it when: Comparing areas of object and image.

Differentiating a power

ddx(xⁿ) = n xⁿ⁻¹

Use it when: Finding a gradient function.

Gradient at a point

substitute the x value into dydx

Use it when: The gradient of a curve at one particular point.

Turning points

solve dydx= 0

Use it when: Maximum or minimum of a curve.

Integrating a power

∫xⁿ dx = xⁿ⁺¹/(n + 1) + c

Use it when: Reversing differentiation — never forget + c.

Only fails when n = −1.

Magnitude

|v| = √(x² + y²)

Use it when: Length of a vector (x; y).

Vector between two points

AB = OB − OA

Use it when: Working from position vectors.

Midpoint of AB

OM = 12(OA + OB)

Use it when: M is halfway along AB.