Pure Mathematics

WJEC A-Level Mathematics

How to solve simultaneous equations?

How to solve simultaneous equations?

1. rearrange linear equation

2. set equal to one another

3. find value and sub back in

4. find other value

How to complete the square?

How to find the min/max of a value?

1. complete the square

2. sketch graph

How to solve quadratic inequalities?

1. solve normally

2. state critical values

3. sketch graph to visualise

4. shade region by looking at inequality

5. state inequalities range

Discriminant when 2 real roots

b2-4ac > 0

Discriminant when 1 real root

b²-4ac = 0

Discriminant when no real roots

b²-4ac < 0

Distance between 2 points

Midpoint of a line

Gradient of a straight line

Equation of a straight line passing 2 points

How to know if lines are perpendicular?

if m₁ x m₂ = -1

When are lines parallel?

If they have the same gradient

How to find intersection of lines?

solve equations simultaneously

Equation of circle

General form of equation of circle

How to find equation of a tangent to a circle?

1. find gradient of radius

2. find gradient of tangent (perpendicular)

3. sub into equation of line using point co-ordinate

Point outside circle

CP > radius

Point inside circle

CP < radius

Point on circle

CP = radius

Circles touch externally

C₁C₂ = R₁₊R₂

Circles do not touch

C₁C₂ > R₁₊R₂

Circles intersect at 2 points

C₁C₂ < R₁₊R₂

Circles touch internally

C₁C₂ = R₁—R₂

Line intersects at 2 points

chord

Line intersects at 1 point

tangent

How to differentiate?

1. multiply by power

2. minus 1 from power

3. gradient function

Differentiation from first principles

Gradient of normal

perpendicular to gradient of tangent

How to integrate?

1. gradient function

2. increase power by 1

3. divide by new power

4. must have +c

Quadratic formula

a^m/n

ⁿ√a^m

a^m x aⁿ

a^m+n

a⁰

1

a^-n

1/aⁿ

(a^m)ⁿ

a^mn

a^m / aⁿ

a^m-n