# Pure Core Equations

AQA Maths A-Level

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## Pages in this set

### Page 1

Pure Core
Indices:
xm×xn = xm+n

xm = xm-n
xn

n
xm = xmn

x0 = 1 (x0)

x-m = x1
m

m n
x n = xm

Logarithms:
x = mn then logm x = n

log ab = log a + log b

log a
b = log…

### Page 2

Exponentials and Logarithms:
dy
y = ekx then dx = kekx

ekx = 1 kx
ke + c

y = ln x then dy
dx =
1
x

1
x = ln x + c

ff((xx)) = ln f (x) + c (f (x) > 0)

y = ax is…

### Page 3

y = ax2 + bx + c (a0)

Difference of two squares = a2 - b2 = (a - b)(a + b)

C ompleting the square is y = (x + p)2 + q where p = b
2

V ertex of the graph where x + p…

### Page 4

Functions:
xER means x is a real number.

Domain is the set of permitted values for x

Range is the set of permitted values for y

|x| = x for x0 and - x for x < 0

gf (x) = g[f (x)]

Partial Fractions:
Polynomial of degree n
Polynomial…

### Page 5

Coordinate geometry:
Distance between points A(x1, y1)and B(x2, y2) =
(x - x )
2 1
2
+ (y2 - y1)2
y2-y1
Gradient of straight line joining A(x1, y1)and B(x2, y2) = x2-x1

x1+x2 y1+y2
M id - point of straight line joining A(x1, y1)and B(x2, y2) = ( 2…

### Page 6

Trigonometry:
sin x = O
H

A
cos x = H

tan x = O
A

Sine rule is sina b
A = sin B =
c
sin C

I nverted sine rule is sina
A = sin B =
b
sin C
c

C osine rule is c2 =…

### Page 7

cos 3x = 4(cos x) 3 - 3cos x

3
tan 3x =3tan x-(tan x)
1-3(tan x )2

Rsin (x±) or Rcos (x±) is the harmonic form (R > 0 and is acute)

### Page 8

Dif f erentiation :

F inds the gradient of a f unction.

dy n
dxax = anxn-1

d2y
dx2
= dy dy
dx[ dxf (x)]

2
M aximum point when d y
dx2
<0

2
M inimum point when d y
dx2
>0

dy
dx[f (x)±g(x)] = dy dy dy…

### Page 9

b
3 0 n
rule = yh [y + y + 4(y + y ... + y
Simpsons 1 3 n-1) + 2(y2...yn-2)]
a

where h = b-a
n and n is even

### Page 10

V ectors:

b) = |v| = a + b AB + BC = AC u = kv when u is parallel to v
2
M agnitude of v(a 2

P osition vector of P = OP M id - point of AB = 1
2(a + b) AB is direction…