OCR Additional Maths

Tanθ = (Trig Identity)

sinθ ÷ cosθ

1 = (Trig Identity)

sin²θ + cos²θ

a² = (Cosine Rule)

b² + c² -2bccosA

cosA = (Cosine Rule)

(b² + c² -a²) ÷ (2bc)

Remainder of f(x) when divided by (x-a)

f(a) - substitute in a to get remainder

When is (x-a) a factor of f(x)

f(a) = 0 and x=a is a root

degree/order of polynomial

highest power

Binomial Expansion

(a+b)² = nC0aⁿ + nC1aⁿ-¹b ... + nCnbⁿ

nCr =

n! ÷ (r!(n-r)!)

Binomial Distribution

P(X=r) = nCrp^rq^(n-r)

X has the binomial distribution for

X~B(n,p) where n= number of trials and p= probability of success in each trial

equation of a tangent

y - y₁ = m(x - x₁)

gradient of a line, m

(y₁-y₂) ÷ (x₁-x₂)

m=

tanθ

Length of a line

√(x₁-x₂)²+(y₁-y₂)²

Formula of a circle, centre (0,0)

x² + y² = r²

Formula of a circle, centre (a,b)

(x-a)² + (y-b)² = r²

Sine Rule

(a÷sinA) = (b÷sinB)

Differentiation (axⁿ)

naxⁿ⁻¹

gradient of y

dy ÷ dx

gradient of f(x)

f`(x)

Stationary Points

Where (dy÷dx) = 0

Integration

∫xⁿ dx = (xⁿ+¹)÷ (n+1) +c

Area under a curve

∫ba f`(x)dx = [f(x)]ba = f(b) - f(a)

Area between two curves

∫f(x)-g(x)dx where f(x) is the top curve and g(x) is the bottom curve

s= (Know a)

ut + ½at²

s = (Know v)

((u+v)÷2)×t

v =

u + at

v²=

u² + 2as

s=

vt - ½at²

displacement → velocity → acceleration

differentiation

acceleration → velocity → displacement

integration

Factorial

x!

Permutations

Order matters, where n = number of total choices and r = number of items being chosen

Combination

Order is not important, nCr

log(xy)

log(x) + log(y)

log(x/y)

log(x) - log(y)

log(xⁿ)

n log(x)

log(ⁿ√x)

¹/n log(x)

log(1)

0

log(¹/x)

log(1) - log(x) = -log(x)

log(10)

1

y = ka^x

log(y) = log(k) + x log(a)

where on a graph of log(y) against x, m = log(a) and y-intercept = (0, log(k))

y = axⁿ

log(y) = log(a) + n log(x)

where on a graph of log(y) against log(x), m = n and y-intercept = (0, log(a))

trapezium rule area

1/2(sum of parallel sides) x width

midpoint formula

(x₁+x₂)÷2 , (y₁+y₂)÷2

perpendicular gradient

-1/m

discriminant

b²−4ac

two distinct roots

b²−4ac > 0

repeated root

b²−4ac = 0

no real roots/intersection

b²−4ac < 0

if tangent, look for

repeated root

so b²−4ac = 0

tan graph

180 +/-

(if tanx = +, then 180 +)

sin graph

180-

360+

cos graph

360-