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Math > pure > Flashcards

pure Flashcards

1
Q

new formula of a line

A

y-y₁ = m(x-x₁)

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2
Q

gradient of a straight line through 2 point

A

change in y / change in x

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3
Q

midpoint of a line

A

(x₁+x₂/2 , y₁+y₂/2)

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4
Q

quadratic formula

A

-b ∓ √b²-4ac / 2a

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5
Q

set notation for and inequality

A

{x: x> -3} ∩ {x: x<2}

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6
Q

set notation for or inequality

A

{x: x<-5} ∪ {x: x>1}

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7
Q

discriminant

A

b² - 4ac

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8
Q

discriminant when 2 different real roots

A

b² - 4ac > 0

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9
Q

discriminant when no real roots

A

b² - 4ac < 0

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10
Q

discriminant when 2 equal/repeated roots

A

b² - 4ac = 0

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11
Q

general form of a circle with centre (a, b)

A

(x-a)² + (y-b)² = r²

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12
Q

gradient of normal to a point on a circle

A

gradient from centre to radius

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13
Q

gradient of tangent to a point on a circle

A

negative reciprocal of normal gradient

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14
Q

the angle in a semicircle is…

A

a right angle

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15
Q

a line from the centre of a circle to a chord, at a right angle to that chord will…

A

bisect the chord and is the shortest distance from the chord to the centre

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16
Q

tangents from the circle to the same point outside the circle…

A

have the same length

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17
Q

(x+3) is a factor of 6x³-bx² + 18, find the value of b.

A

substitute x=-3

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18
Q

f(x) → -f(x)

A

reflection in x axis

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19
Q

f(x) → f(-x)

A

reflection in y axis

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20
Q

sine rule

A

a/sinA = b/sinB = c/sinC

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21
Q

cosine rule

A

a² = b² + c² - 2bc cosA

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22
Q

area of a triangle trig

A

0.5 × a × b × sinC

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23
Q

sin graph repetitions

A
  • add 360
  • 180 minus x
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24
Q

cos graph repetitions

A
  • add 360
  • minus x plus 360
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25
Q

tan graph repetitions

A
  • plus 180
  • minus 180
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26
Q

cos²x + sin²x

A

1

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27
Q

tan(x)

A

sin(x) / cos(x)

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28
Q

logₐa

A

1

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29
Q

logₐ1

A

0

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30
Q

logₐ(1/a)

A

-1

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31
Q

logₐx + logₐy

A

logₐ(xy)

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32
Q

logₐx - logₐy

A

logₐ(x/y)

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33
Q

logₐ(xⁿ)

A

nlogₐx

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34
Q

logₐ(aˣ)

A

x

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35
Q

general equation for exponential growth

A

V = Akᵗ

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36
Q

y = axⁿ

A

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

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37
Q

y = kbˣ

A

log(y) = log(k) + xlog(b)

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38
Q

when to use (nCr)

A

(a+b)ⁿ
where n is a positive whole number

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39
Q

when to use n(n-1)/2!

A

(1+x)ⁿ
where n is a rational number

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40
Q

differentiate eⁿˣ

A

neⁿˣ

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41
Q

gradient of perpendicular

A

negative reciprocal

42
Q

when is a function increasing

A

gradient is positive

43
Q

when is a function decreasing

A

gradient is negative

44
Q

show that the square of any number is positive in symbols

A

x² ≥ 0

45
Q

gradient at stationary point

A

0

46
Q

second derivative at minimum point

A

> 0

47
Q

second derivative at maximum point

A

< 0

48
Q

integration steps

A
  • add 1 to power
  • divide by power
49
Q

method to integrate 3x² if limit is unknown

A

factorise 3

50
Q

vertical line test

A

If no vertical line will ever cross the graph more than once, it is a function.

51
Q

horizontal line test

A

If no horizontal line will ever cross the graph more than once it is a one-to-one function, else many-to-one.

52
Q

ℕ

A

natural numbers: all positive whole numbers

53
Q

ℤ

A

integers: all whole numbers

54
Q

ℚ

A

rational numbers: all numbers that can be written as an integer fraction

55
Q

ℝ

A

real numbers: all possible decimals

56
Q

transformation from f(x) to f⁻¹(x)

A

reflection in line y=x

57
Q

domain of inverse is

A

range of original

58
Q

ff⁻¹(x)

A

x

59
Q

how many radians is 360º

A

2π

60
Q

1ᶜ

A

angle subtended at the centre of the circle by an arc equal in length to the radius

61
Q

length of an arc of a circle

A

l = rθ

62
Q

area of a sector of a circle with radians

A

A = ½r²θ

63
Q

tan²x + 1

A

sec²x

64
Q

cot²x + 1

A

cosec²x

65
Q

what does tan(arcsin¾) mean

A

find tanθ if sinθ=¾

66
Q

sin2θ

A

2sinθcosθ

67
Q

cos2θ

A

cos²θ - sin²θ

68
Q

tan2θ

A

2tanθ / 1-tan²θ

69
Q

asinθ - bcosθ

A

sin(θ - α)

70
Q

acosθ - bsinθ

A

cos(θ + α)

71
Q

if both transformations are in the x direction…

A

do translation first

72
Q

how to describe transformations for a quadratic

A

complete the square

73
Q

differentiate sin(x)

A

cos(x)

74
Q

differentiate cos(x)

A

-sin(x)

75
Q

steps of chain rule

A
  • bring down exponent
  • multiply derivative of bracket
  • reduce exponent by 1
  • keep bracket the same
76
Q

differentiate ln(x)

A

1/x

77
Q

how to differentiate ln(…)

A

derivative of bracket / bracket

78
Q

differentiate aᵏˣ

A

k(ln(a))aᵏˣ

79
Q

when to use product rule

A

when there is more than 1 unknown

80
Q

differentiate y

A

dy/dx

81
Q

how to determine whether concave

A

2nd derivative < 0

82
Q

how to determine whether convex

A

2nd derivative > 0

83
Q

point of inflection

A

between concave and convex

84
Q

2nd derivative at point of inflection

A

0

85
Q

how to make an equation equal to zero

A

make numerator 0

86
Q

how to make an equation undefined

A

make denominator 0

87
Q

how to solve an equation with sin ∓ cos

A

harmonic form

88
Q

what type of function does not have an inverse

A

many to one

89
Q

when to integrate using partial fractions

A

when the denominator has more than one x

90
Q

order to choose for u when integrating by parts

A

Ln
Algebra
Trig
Exponentials

91
Q

how to integrate lnx

A

u=lnx
v=1

92
Q

integral of ln(x)

A

xln(x) - x + c

93
Q

increasing sequence

A

If each term of the sequence is greater than the one immediately preceding it.

94
Q

decreasing sequence

A

If each term of the sequence is less than the one immediately preceding it.

95
Q

periodic sequence

A

A sequence that repeats itself at regular intervals.

96
Q

order of a sequence

A

the number of terms before the sequence repeats

97
Q

inductive sequence

A

Defines the term in a sequence based on the previous term.

98
Q

convergent sequence

A

tends towards a limit

99
Q

general rule for an arithmetic sequence

A

Uₙ = a + (n-1)d

100
Q

general rule for a geometric sequence

A

Uₙ = arⁿ⁻¹

101
Q

formula to differentiate parametric equations

A

(dy/dt) / (dx/dt)

102
Q

formula to integrate parametric equations

A

∫y (dx/dt)

Math (5 decks)

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