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Page 1

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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

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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

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Quadratic functions:
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

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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

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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

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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

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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

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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

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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

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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…

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