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A-level Maths > pure core AS > Flashcards

pure core AS Flashcards

1
Q

A^m/n

A

(n√a)^m

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2
Q

discriminant

A

b^2 - 4ac

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3
Q

b^2 - 4ac < 0

A

no real roots

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4
Q

b^2 - 4ac = 0

A

1 real root

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5
Q

b^2 - 4ac > 0

A

2 real roots

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6
Q

dotted line

A

<>

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7
Q

solid line

A

<= >=

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8
Q

[]

A

included in the range

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9
Q

()

A

not included in the range

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10
Q

midpoint

A

( (x1+x2)/2 , (y1+y2)/2)

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11
Q

length of line

A

√(x1-x2)^2 + (y1-y2)^2

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12
Q

the angle in a semi circle is always

A

a right angle

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13
Q

the perpendicular line from the centre of a circle to a chord

A

perpendicularly bisects the chord

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14
Q

the tangent to a circle at a point

A

is perpendicular to the radius through that point

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15
Q

tanx =

A

sinx/cosx

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16
Q

sin^2 x =

A

1 - cos^2 x

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17
Q

y = f(x) + 3

A

translation 3 in the y direction

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18
Q

y = f(x+3)

A

translation -3 in the x direction

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19
Q

y = 3f(x)

A

stretch of scale factor 3 in the y direction

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20
Q

y = f(3x)

A

stretch of scale factor 1/3 in the x direction

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21
Q

y = -f(x)

A

reflection in the x axis

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22
Q

y = f(-x)

A

reflection in the y axis

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23
Q

nCr

A

n! / r!(n-r)!

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24
Q

normal to curve

A

perpendicular to tangent at particular point

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25
Q

vector polar form

A

(r, θ)

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26
Q

vector/component form

A

(X, Y)

Xi + Yj

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27
Q

position vector

A

starts at the origin

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28
Q

unit vector

A

magnitude of 1

divide a direction by its magnitude to get its unit vector

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29
Q

vectors with common multiples

A

are parallel

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30
Q

vectors are perpendicular if

A

their dot/scalar product equals zero

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31
Q

Inverse of

“A to the power of X equals B”

A

log to the base a of b equals x

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32
Q

log(xy) =

A

log(x) + log(y)

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33
Q

log(x/y)

A

log(x)-log(y)

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34
Q

log to the base a of a =

A

1

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35
Q

log1/y =

A

-logy

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36
Q

log1=

A

0

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37
Q

degrees to radians

A

a*π/180

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38
Q

radians to degrees

A

a*180/π

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39
Q

arc length formula degrees

A

ϴ/360 * 2πr

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40
Q

arc length formula radians

A

rϴ

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41
Q

sector area formula degrees

A

ϴ/360 * πr^2

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42
Q

sector area formula radians

A

1/2r^2ϴ

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43
Q

Small angle approximation sine

A

ϴ = sinϴ = tanϴ

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44
Q

small angle approximation cos

A

cosϴ = 1 - 1/2ϴ^2

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45
Q

inputs of a function

A

domain

46
Q

outputs of a function

A

range

47
Q

composite functions

A

work from inside out

48
Q

a composite function has a domain…

A

of the first function

49
Q

for an inverse function to exist

A

the function must be one-to-one

50
Q

geometric relationship between inverse function and function

A

reflected in line y=x (domain and range swap)

51
Q

period of a sequence

A

how often the sequence repeats

52
Q

increasing sequence

A

every term is greater than the previous term

53
Q

decreasing sequence

A

every term is less than the previous term

54
Q

diverging sequence

A

the difference between each term gets greater away from a point

55
Q

converging sequence

A

the difference between each term gets less towards a point

56
Q

sum of an arithmetic sequence

A

s = 1/2n (a+l)

57
Q

term in an arithmetic sequence (last term)

A

l = a + (n-1)d

58
Q

example of a geometric sequence

A

a, ar, ar^2, ar^3, ar^4…

59
Q

sum of terms in a geometric sequence

A

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

60
Q

what happens if the common ratio is between -1 and 1

A

r^n tends to 0 and n tend to infinity
so s = a/(1-r)
series converges and has a sum to infinity

61
Q

sketching y=|f(x)|

A

swap all the negative y values to positive y values

62
Q

sketching y =f(|x|)

A

get rid of -x graph and mirror positive x in the y axis

63
Q

Convex curve

A

Curve underneath chord

F’‘(X)>0

64
Q

Concave curve

A

Curve above chord

F’‘(X)<0

65
Q

chain rule

A

dy/dx = dy/du * du/dx

66
Q

Product rule

A

Vdu+Udv

67
Q

Quotient rule

A

(Vdu- udv )/v^2

68
Q

Sin cos identity

A

Sin^2x+cos^2x = 1

69
Q

Tan sin cos identity

A

Tanx = sinx/cosx

70
Q

Tan sec identity

A

Tan^x + 1 = sec^2x

71
Q

Cot cosec identity

A

1+cot^2x = cosec^2x

72
Q

compound angle formulae sin

A

sin(θ+ϕ) = sinθcosϕ + sinϕcosθ

73
Q

compound angle formulae cos

A

cos(θ+ϕ) = cosθcosϕ - sinθsinϕ

74
Q

compound angle formulae tan

A

tan(θ+ϕ) = (tanθ+tanϕ)/(1-tanθtanϕ)

75
Q

Double angle formulae sin

A

sin(2θ) = 2sinθcosθ

76
Q

Double angle formulae cos

A

cos(2θ)= cos^2(θ) - sin^2(θ)

77
Q

Double Angle formulae tan

A

tan(2θ) = 2tanθ / 1- tan^2(θ)

78
Q

compound angle formulae proof sin

A

triangle with obtuse angle at C

area of ABC = area of ADC + area of DBC

79
Q

compound angle formulae proof cos

A

let θ = 90-1 in sin compound formula

80
Q

d/dx lnx

A

1/x

81
Q

d/dx a^x

A

a^x*lna

82
Q

d/dx e^x

A

e^x

83
Q

d/dx cosx

A

-sinx

84
Q

d/dx sinx

A

cosx

85
Q

d/dx tanx

A

sec^2x

86
Q

d/dx cotx

A

-cosec^x

87
Q

d/dx secx

A

sec xtanx

88
Q

d/dx cosecx

A

-cosecxcotx

89
Q

differentiating implicitly

A
  • differentiate with respect to x
  • if y term differentiate and multiple by dy/dx
  • if xy product use product rule and multiply by dy/dx
  • factor out the dy/dx
90
Q

turning point from implicit differentiation

A

solve simultaneously with the equation of the curve

91
Q

∫f’(x)/f’‘(x) =

A

ln|f(x)| + c

92
Q

∫sinkx dx

A

-1/k coskx +c

93
Q

∫coskx dx

A

1/k sinkx + c

94
Q

∫ sec^2 kx dx

A

1/k tan kx + c

95
Q

∫e^kx dx

A

1/ke^kx + c

96
Q

∫1/x dx

A

ln|x| + c

97
Q

parametric -> cartesian method 1

A
  • rearrange one equation in terms of the parameter

- substitute into the second equation

98
Q

parametric -> cartesian method 2

A
  • add the equations
  • rearrange this for the parameter
  • substitute into either equation
99
Q

parametric equations for a circle (centre (a,b))

A 
x = a + rcosθ
y = b + rsinθ
100
Q

parametric differentiation (parameter t)

A

dy/dx = (dy/dt) / (dx/dt)

101
Q

parametric integration

A

∫ydx = ∫y* (dx/dt) dt

102
Q

change of sign methods

A

find a solution to f(x) by finding the range where the function changes sign

103
Q

drawbacks of change of sign methods

A
  • if the curve touches the axis it won’t work
  • if roots are close together it is easy to miss changes in sign
  • if there is discontinuity in the function the method may say there are false roots
104
Q

fixed point iteration

A

rearrange the function into the form x = g(x)

input values of x into g(x) to find the next value of x until the outputted value is repeating

105
Q

what does the iterative formula do?

A

instead of looking for where y=f(x) crosses the x axis look for where y=g(x) intersects with the line y=x

106
Q

drawing spirals / staircases

A
  • draw y = g(x) and y=x
  • mark this initial estimate
  • draw a vertical line to the curve
  • draw a horizontal line to y=x, mark the x value
  • repeat atleast 5 times
107
Q

Newton Raphson Method

A
  • draw a tangent to the curve
  • use a triangle to find the gradient of the tangent, this can be written as f’(x)
    f(x1)/ x1 - x2
  • rearrange for the estimated root value (x2)
  • iterate over to find the expected root
108
Q

Trapezium rule

A

A = 1/2h [(y0 + yn) + 2(y1 + y2 + …. + yn-1)

109
Q

trapezium rule overestimate

A

convex curve

110
Q

trapezium rule underestimate

A

concave curve

111
Q

upper/lower bounds for trapezium rule

A

use square with different corners touching the curve

A-level Maths (6 decks)

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