Pure Core Equations

AQA Maths A-Level

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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 a - log b
log an = nlog a
logm m = 1
y = ax then x = loga y
Expenential function is a function in the form y = ax

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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 exponential growth
y = a-x is exponential decay
e0 = 1
Surds (irrational numbers):
a×b = ab
a
b
= a
b
ac±bc = (a±b)c
Rationalise the fraction a±1 by multiply by ab
b ab…read more

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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 = 0
x = -b±2
2
b -4ac
a
b2 - 4ac is the discriminant
b2 - 4ac > 0 means there are 2 distinct roots
b2 - 4ac = 0 means there is one…read more

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Functions:
xER means x is a real number.…read more

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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 , 2 )
Gradient of perpendicular line = -1
m
C artesian equation is y = mx + c
y - y1 = m(x - x1)
ax + by + c = 0
Equation of…read more

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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 = a2 + b2 - 2abcos C
2 2 2
I nverted cosine rule is cos A = b +2
c -a
bc
Area of a triangle = 1 1
2bh = 2absin C
T o…read more

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

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Dif f erentiation :
F inds the gradient of a f unction.…read more

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Simpsons 1 3 n-1) + 2(y2...…read more

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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 vector from a to b
Scalar product = ab = |a||b|cos where is the angle between the vectors ab = a1b1 + a2b2
ab = 0 means the vectors are perpendicular
r = a +…read more

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