IGCSE Maths formulae & stuff to remember

{ }

the set of

n(A)

number of elements in set A

∈

is an element of

∉

is not an element of

∅

the empty (null) set

ξ

the universal set

∪

union - everything in both sets - everything that is in at least one of the sets

∩

intersection - the overlap of the sets - elements that are in both sets

⊂

is a subset of

A′

the complement of A - everything in the universal set not in A

open interval on a number line - number isn’t included - < or >

closed interval on a number line - number is included - ≥ or ≤

area of a pyramid

1/3 x base area x height

area of a circle

𝜋r²

circumference of a circle

2𝜋r

area of a parallelogram

base x vertical height

area of a triangle

½ x base x height

area of a sector of a circle

angle x 𝜋r²
360

circumference of a sector of a circle

angle x 2𝜋r
360

area of a rhombus

width x height (diagonals)
2

probability of an event when all outcomes are equally likely

number of ways the event can happen / total number of possible outcomes

mutually exclusive events

events that can’t happen at the same time eg rolling a 1 & 3 on a dice

independent events

events that have no effect on the probability of each other

p(event A and event B)

p(A) x p(B)

p(event A or event B)

p(A) + p(B)

dependent/conditional events

events that depend on/are influenced by other events. opposite of independent events

line formula

y = mx + c

m = gradient

c = y-intercept

quadratic graphs

parabola - u shape

symmetric with 1 turning point

positive x² = u-shape, negative x² = n shape

find roots (points where it crosses x axis) by solving

cubic graphs

curve with a wiggle

2 turning points

positive x³ = curve goes down → up (left → right)

last number of the equation tells you height

reciprocal graphs

a
x

recognise them by their fraction

the denominators never = 0 (function is undefined if so). asymptotes are lines that the graph gets very close to but never touches

if the term is squared it will only have positive values so will only be in the top 2 quarters

exponential graphs

y = a^x

when a > 1 the graph increases really quickly. when a < 1 the graph decreases really quickly

always go through (0,1)

circle graphs

(x - a)² + (y - b)² = r²

r = radius

centre is (a,b) but remember to swap the signs

1 tonne in kg

1000kg

1ml in cm³

1cm³

which way are bearings measured from

clockwise from the north

what does ‘bearing of b from a’ mean

start at a go to b

how to set bearing out

3 digits eg 045 degrees

x = x

opposite angles are equal

corresponding angles are equal

look for the f shape

alternate angles are equal

look for the z shape

co-interior angles add to 180 degrees

the ambiguous case

if an angle is supposed to be obtuse but you’re getting an acute one (or vice versa) subtract what you’re getting from 180 degrees

simultaneous equations acronyms

SSS = same signs subtract

DSA = different signs add

>

greater than

<

less than

≥

greater than or equal to

≤

less than or equal to

what happens when you multiply/divide both sides of an inequality by a negative number

flip the inequality sign

line used for < or > when graphing inequalities

dashed/dotted line

line used for ≥ or ≤ when graphing inequalities

soild line

tree diagram rules

across diagram = multiply

down diagram = add

LSF, ASF, VSf

ASF = LSF²

VSF = LSF³

parts of a circle

angle at centre is 2x angle at circumference - arrowhead

angle at circumference in a triangle on the diameter is 90^o

angles in the same segment are equal - butterfly

remember to stay in the same segment

opposite angles in a cyclic quadrilateral add to 180^o

angle made by tangent & radius is 90^o

alternate segment theorem

look for a triangle / chord meeting a tangent

2 tangents coming from the same point are the same length

often leads to an isosceles triangle

how many mg in a g

1000

compound interest formula

original amount x multiplier^years

eg £300 increase by 5% for 2 years = 300 × 1.05²

frequency density formula

frequency / class width

density formula

density = mass / volume

ρ = m/V

nth term of an arithmetic sequence

u_n = a + (n - 1) d

y = sinx graph

look for sideways S shape

y = cosx graph

look for sideways C shape

y = tanx graph

how to find LCM from prime factors

find the highest power of EVERY prime (doesn’t have to be common) & multiply them together

how to find HCF from prime factors

find the highest power of every prime that APPEARS IN BOTH (must be common) & multiply them together

y = f(x) + a

translation up/down (vector ⁰_a) - affects the y value

eg f(x) + 5 the point 2,3 goes to 2,8

y = f(x + a)

translation left/right (vector ^-a₀) - affects the x value - OPPOSITE SIGN

eg f(x + 5) the point 7,3 goes to 2,3

y = -f(x)

reflection in the x-axis - affects the y value

eg point 2,3 goes to 2,-3

y = f(-x)

reflection in the y-axis - effects the x value

eg point 2,3 goes to -2,3

y = af(x)

stretch by scale factor a in the y-direction (effects y value)

eg for 5f(x) the point 2,3 goes to 2,15

y = f(ax)

stretch by scale factor 1/a in x-direction (effects x value)

eg for f(3x) the point 6,7 goes to 2,7

intersecting chord theorem - 2 chords intersect inside the circle

circumference→middle x middle→circumference = other circumference→middle = middle→circumference

intersecting chord theorem - 2 chords intersect outside circle

whole one x outside = whole one x outside

intersecting chord theorem - chord and tangent meet outside circle

tangent² = whole thing x outside

kinematics order

differentiate: displacement → velocity → acceleration

percentage change

(change/original) x 100

completing the square

a^m x aⁿ

a^{m + n}

a^m / aⁿ

a^m-n

(a^m)ⁿ

a^mn

a^-n

1
aⁿ

a^1/n

gradient formula

aka rise/run or change in y / change in x

midpoint of a line

sum of interior angles in a polygon

(n-2) x 180

sum of exterior angles in a polygon

always 360

volume of a prism

cross sectional area x length

pressure

force / area

finding median/quartiles for odd vs even number of terms in a list

Odd number of data values = IGNORE the median when splitting (all numbers except median either on one side or the other) eg:

Even number of data values = Split straight down the middle (all numbers either on one side or the other)

to find quartiles find the median of each other two new subsets, it doesn’t matter if it isn’t a number in the original list & can be a decimal
eg here the LQ would be 4.5 and the UQ would be 14 (neither on the list)