STEP formula

Question 1

What is the quadratic formula

Answer 1

(-b±Root(b^2-4ac))/2a

Question 2

formula for coefficients of ax^2 + bx + c = 0 in terms of roots

Answer 2

α + β = −b/a

αβ = c/a

Question 3

formula for coefficients of ax^3 + bx^2 + cx + d = 0

Answer 3

α + β + γ = −b/a,
αβ + βγ + γα = c/a,
αβγ = −d/a

Question 4

4 indices laws

Answer 4

a^x*a^y=a^(x+y)
a^0=1
(a^x)^y=a^(xy)
a^x=e^(xlna)

Question 5

4 log rules

Answer 5

x=a^n <=> n=loga(x)
loga(x)+loga(y) = loga(xy)
loga(x)-loga(y) = loga(x/y)
kloga(x)=loga(x^k)

Question 6

nth term of arithmetic series

Answer 6

u(n) = a + (n − 1)d

a is initial n is number of terms, d is difference

Question 7

nth term of geometric

Answer 7

ar^(n-1)

a is initial n is number of terms

Question 8

sum of arithmetic

Answer 8

S(n) = 1/2(n{2a + (n − 1)d})

a is initial n is number of terms, d is difference

Question 9

sum of geometric

Answer 9

(a(1-r^n))/(1-r)

Question 10

limit of sum of geometric to infinity

Answer 10

a/(1-r)

mod(r) <1

Question 11

nCr

Answer 11

n!/(r!(n-r)!)

Question 12

(a+b)^n

Answer 12

sum (from r=0 to n) of nCr*a^(n-r)b^r

Question 13

(1+x)^k

Answer 13

1+kx+k(k-1)/2! * x^2 + k(k-1)(k-2)/3! *x^3 …+ k(k-1)…(k-r+1)/r! x^r +…
mod(x) <1 to converge

Question 14

sum of natural numbers
sum of (from r=1 to n) of r

Answer 14

1/2*n(n+1)

Question 15

maclaurin series

Answer 15

f(x) = sum of (from r=0 to infinity) 1/r! f^r(0)x^r

Question 16

maclaurin e^x

Answer 16

= sum of (from r=0 to infinity) (x^r)/r!

Question 17

maclaurin ln(1+x)

Answer 17

= sum of (from r=0 to infinity) (-1)^(r+1) * (x^r)/r

x -x^2/2 +x^3/3 …

Question 18

maclaurin sinx

Answer 18

= sum of (from r=0 to infinity) (-1)^(r) * (x^(2r+1))/(2r+1)!

Question 19

maclaruin cosx

Answer 19

= sum of (from r=0 to infinity) (-1)^(r) * (x^(2r))/(2r)!

Question 20

which maclaurin converge

Answer 20

sinx,cosx e^x converge for all x

ln(1+x) converges for -1< (x) <=1

Question 21

Straight line through point (x1,y1) and gradient m

Answer 21

y-y1 = m(x-x1)

Question 22

perpendicular condition

Answer 22

m1m2 = -1

Question 23

sine rule

Answer 23

a/sinA = b/sinB = c/sinC

Question 24

cosine rule

Answer 24

a^2 = b^2 + c^2 − 2bc cos A

Question 25

area of a triangle

Answer 25

1/2 ab sin C

Question 26

trig pythag identity

Answer 26

cos^2 A + sin^2 A = 1

Question 27

trig pythag tan

Answer 27

1 +tan^2 A = sec^2 A

Question 28

trig pythag cot

Answer 28

cot^2 A +1 = cosec^2 A

Question 29

sine double angle

Answer 29

sin A+-B = sin A cos B +_ sin B cos A

Question 30

cosine double angle

Answer 30

cos A+-B = cos A cos B -+ sin A sin B

Question 31

tan double angle

Answer 31

tan (A+B) = (tan A +- tan B)/(1 -+ tan A tan B)

Question 32

small angle approximations

Answer 32

sin θ ≈ θ , cos θ ≈ 1 − 1/2 θ^2 , tan θ ≈ θ | θ in radians and small

Question 33

sinhx

Answer 33

(e^x-e^-x)/2

Question 34

coshx

Answer 34

(e^x+e^-x)/2

Question 35

tanhx definition

Answer 35

= sinhx/coshx

Question 36

hyperbolic trig pythag

Answer 36

cosh^2 A − sinh^2 A = 1

Question 37

pythag sech A

Answer 37

1-tanh^2 A = sech^2 A

Question 38

pythag cosech A

Answer 38

- coth^2 A +1 = -cosech^2 A | cosech^2A = coth^2 A -1

Question 39

sinh double angle

Answer 39

sinh(A ± B) = sinh A cosh B ± cosh A sinh B

Question 40

cosh double angle

Answer 40

cosh(A ± B) = cosh A cosh B ± sinh A sinh B

Question 41

tanh double angle

Answer 41

tanh(A ± B) = (tanh A ± tanh B)/(1 ± tanh A tanh B)

Question 42

d/dx sinx

Answer 42

cosx

Question 43

d/dx cos x

Answer 43

-sinx

Question 44

d/dx tanx

Answer 44

sec^2 x

Question 45

d/dx cot x

Answer 45

-cosec^2 x

Question 46

d/dx cosecx

Answer 46

-cosec x cotx

Question 47

d/dx sec x

Answer 47

sec x tan x`

Question 48

d/dx arcsin x

Answer 48

1/root(1-x^2)

Question 49

d/dx arctan

Answer 49

1/(1+x^2)

Question 50

why isnt arcos x included

Answer 50

bc arcos x = 1/2 π − arcsin x

Question 51

d/dx sinhx

Answer 51

coshx

Question 52

d/dx coshx

Answer 52

sinhx

Question 53

d/dx tanhx

Answer 53

sech^2 x

Question 54

d/dx cothx

Answer 54

- cosech^2x

Question 55

d/dx sechx

Answer 55

sechx tanhx

Question 56

d/dx arsinh x

Answer 56

1/(root(1+x^2)

Question 57

d/dx tanhx

Answer 57

1/(1-x^2)

Question 58

d/dx e^x

Answer 58

e^x

Question 59

product rule

Answer 59

ab'+a'b

Question 60

chain rule

Answer 60

d/dx f(g(x))= g'(x)f'(g(x))

Question 61

quotient rule

Answer 61

d/dx u/v = (vu'-uv')/v^2

Question 62

integral x^-1

Answer 62

ln|x| +c

Question 63

what do all integrals need but i cba to write

Answer 63

constants

Question 64

integral x^n

Answer 64

1/(n+1) x^(n+1)

Question 65

integral cos x

Answer 65

sin x

Question 66

integral sinx

Answer 66

-cos x

Question 67

integral sinhx

Answer 67

coshx

Question 68

integral coshx

Answer 68

sinhx

Question 69

integral 1/root(a^2-x^2)

Answer 69

arsin x/a

Question 70

integral 1/(a^2+x^2)

Answer 70

1/a arctan x/a

Question 71

integral e^x

Answer 71

e^x

Question 72

d/dx arcosh x

Answer 72

1/root(x^2-1)

Question 73

integral 1/root(x^2-a^2)

Answer 73

arcosh x/a

Question 74

integral 1/(a^2-x^2)

Answer 74

1/a artanh x/a

Question 75

intergral 1/(x^2-a^2)

Answer 75

either partial fractions or | 1/2a ln| (x-a)/(x+a) |

Question 76

integration by parts

Answer 76

integral of uv' = uv - integral of vu'

Question 77

first principles derivatives

Answer 77

lim (h--> infinty) = (f(x+h)-f(x))/h

Question 78

parametric derivatives

Answer 78

dy/dx = dy/dt / dx/dt

Question 79

volume of rev about x axis

Answer 79

π integral y^2 dx

Question 80

volume of revolution about y axis

Answer 80

π integral x^2 dy

Question 81

trapezium rule

Answer 81

(1/2 h)(y0 + yn + h(y1 + y2 + ··· + yn−1)) | h = (b − a)/n , yr = y(a + rh)

Question 82

shm equation and solution

Answer 82

x¨ = −ω^2 x ⇒ x = R sin(ωt + α) | or x = R cos(ωt + β) or x = A cos ωt + B sin ωt

Question 83

arc length of circle

Answer 83

rθ

Question 84

area of a circle radians

Answer 84

1/2 r^2 θ

Question 85

eulers identity

Answer 85

e^(iθ) = cos θ + i sin θ

Question 86

de moivres theorem

Answer 86

z = r(cos θ + i sin θ) ⇒ z^n = r^n(cos nθ + i sin nθ)

Question 87

roots of unity

Answer 87

z^n = 1 has roots z = e^(2πki/n)

Question 88

half line

Answer 88

arg(z − a) = θ

Question 89

complex circle locus

Answer 89

|z − a| = r

Question 90

the magnitude of a vector

Answer 90

|xi + yj + zk| = | root (x^2 + y^2 + z^2)

Question 91

dot product

Answer 91

a.b = a1b1 + a2b2 + a3b3 = |a| |b| cos θ

Question 92

vector product

Answer 92

a × b = (a2b3 − a3b2)i + (a3b1 − a1b3)j + (a1b2 − a2b1)k = |a| |b||sin θ|n̂

Question 93

euqation of line vectors

Answer 93

r = a + kb

Question 94

equation of plane

Answer 94

(r − a).n = 0 | or r.n = d

Question 95

det of 2x2

Answer 95

det A = ad − bc

Question 96

det of a 3x3

Answer 96

a(minor) -b(minor) +c(minor

Question 97

inverse of a 2x2

Answer 97

1/det (a) * (d -b) | (-c a)

Question 98

(AB)^-1

Answer 98

B^-1 * A^-1

Question 99

reflection matrix in line y=+-x

Answer 99

(0 +-1) | +-1 0

Question 100

rotation in matrix 2x2

Answer 100

(cos θ -sin θ) (sin θ cos θ) anticlockwise about origin

Question 101

rotation about x axis 3x3

Answer 101

(1 0 0 ) (0 cos θ -sin θ) (0 sin θ cos θ)

Question 102

rotation about y axis 3d

Answer 102

(cos θ 0 sin θ ) (0 1 0 ) (-sin θ 0 cos θ)

Question 103

roation about z axis

Answer 103

(cos θ -sin θ 0) (sin θ cos θ 0) ( 0 0 1 )

Question 104

Reflection in place z=0

Answer 104

(1 0 0) (0 1 0) (0 0 -1)

Question 105

Perpendicular distance from a point to a plane

Answer 105

|n1α +n2β + n3γ + d|/root(n1^2 + n2^2 + n3^2)

Question 106

polar coordinates area of a secor

Answer 106

1/2 intergral of r^2 dθ

Question 107

Sum of square numbers

Answer 107

1/6 n (n+1)(2n+1)

Question 108

Sum of cube numbers

Answer 108

1/4n^2(n+1)^2

Question 109

arsinh in ln

Answer 109

ln(x +root(x^2 + 1))

Question 110

Arcosh in ln

Answer 110

ln(x±root(x^2 - 1))

Question 111

artanh in ln

Answer 111

1/2 ln( (1+x)/(1-x) )

Question 112

change of base log formula

Answer 112

loga x = logb x / logb a

Question 113

newton raphson formula

Answer 113

x{n+1}=x{n}-f(x{n})/{f'(x_{n})

Question 114

Probability addition rule

Answer 114

P(A∪B)=P(A)+P(B)−P(A∩B)

Question 115

Probability multiplication rule

Answer 115

P(A∩B)=P(B)P(A|B)

Question 116

Bayes rule

Answer 116

P(B|A) = (P(B)P(A|B))/P(A)

Question 117

nPr

Answer 117

n!/(n-r)!

Question 118

SHM

Answer 118

x'' = -ω^2 x

Question 119

t=

Answer 119

tan(x/2)

Question 120

t formula sin

Answer 120

sinθ = 2t/(1+t^2)

Question 121

t formula cos

Answer 121

cosθ = (1-t^2)/(1+t^2)

Question 122

t formula tan

Answer 122

tanθ = 2t/(1-t^2)

Question 123

dx = (for t-formula)

Answer 123

dx= (2dt)/(1+t^2)

Question 124

Variance

Answer 124

E(X^2) - (E(X))^2

Question 125

Expection

Answer 125

``` E(X) = sum xi * P(X=xi) E(X) = integral xf(x) dx ```

Question 126

E(X^2)

Answer 126

``` E(X^2) = sum (xi)^2 * P(X=xi) E(X) = integral x^2 f(x) dx ```

Question 127

E(aX + bY +c )

Answer 127

aE(X) + bE(Y) +c

Question 128

Var(aX+b)

Answer 128

a^2Var(x)

Question 129

Var(aX+bY+c)

Answer 129

If independent | a^2Var(x)+ b^2Var(x)

Question 130

Binomial

Answer 130

(n/x)p^x(1-p)^x E(x) = np Var(x) = np(1-p)

Question 131

Uniform distribution discrete

Answer 131

1/n | E(X) = 1/2 (n+1)

Question 132

Poisson

Answer 132

lambda^x e^-x /x! | E(X) = Var(X) = lambda

Question 133

Continuous uniform

Answer 133

1/b-a | E(x) = 1/2 (a+b)

Question 134

Normal

Answer 134

``` E(x) = mu Var(x) = sigma^2 ```

Question 135

Independent random variables

Answer 135

P(X=x, Y=y) = P(X=x)P(Y=y)

Question 136

Discrete random variables

Answer 136

P(X=x, Y=y) | => P(X=x) = sum of y from 1 to n of f(x,y)

Question 137

Mutually exclusive

Answer 137

``` P(AUB) = P(A) + P(B) P(AnB) = 0 ```

Question 138

Independent

Answer 138

P(AnB) = P(A)P(B)

Question 139

P(AUB) =

Answer 139

P(A) + P(B) + P(AnB)

Question 140

P(A|B)

Answer 140

P(AnB)/P(B)