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Maths > Pure Maths > Flashcards

Pure Maths Flashcards

1
Q

Pythagorean triples

A

4,3. 5
12,5. 13
24,7. 25
15,8. 17

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2
Q

Sin(A)=

A

Cos(90-A)

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3
Q

Tan θ

A

sin θ / cos θ

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4
Q

Sec x

A

1/cos x

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5
Q

Cosec x

A

1/sin x

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6
Q

Cot x

A

1 / tan x = cos x / sin x = cosec x / sec x

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7
Q

Y = sec θ (graph)

A
  • symmetry in y-axis
  • has period 360
  • Asymptotes of -90, 90, 270
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8
Q

Y = sec θ (graph domain and range)

A
  • domain x∈R, x≠90, 270, 450
  • range y ≤-1 or y≥1
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9
Q

Y = cosec θ (graph)

A
  • has period 360
  • vertical asymptotes for which sin x = 0 (180, 360)
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10
Q

Y = cosec θ (graph domain and range)

A
  • domain x∈R, x ≠ 0, 180, 360
  • range y ≤-1 or y≥1
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11
Q

Y = cot θ (graph)

A
  • has period 180
  • vertical asymptotes for which tan x = 0 (0, 180, 360
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12
Q

Y = cot θ (graph domain and range)

A
  • domain x∈R, x ≠ 0, 180, 360
  • range y∈R,
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13
Q

Sec^2x

A

1 + tan^2 x

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14
Q

Cosec^2x

A

1 + cot^2x

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15
Q

Arcsin x (domain and range)

A

Domain -1≤ x ≤ 1
Range -90 ≤ arcsinx ≤ 90

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16
Q

Arccos x (domain and range)

A

Domain -1≤ x ≤ 1
Range 0 ≤ arccos x ≤ 180

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17
Q

Arctan x (domain and range)

A

Domain x∈R
Range -90 ≤ arctan x ≤ 90

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18
Q

Un (arithmetic)

A

A + (n-1) d

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19
Q

Sn (arithmetic)

A

N/2 (2a + (n-1) d)

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20
Q

Un (geometric)

A

Ar^(n-1)

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21
Q

Sn (geometric)

A

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

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22
Q

Sum to infinity ∞

A

A/1-r
|r| < 1

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23
Q

V=ab^t

A

A= initial value
B= the annual proportional decrease (-ve)/increase (+ve) in the value of the car

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24
Q

Y=log 9 (x+a)

A

(0, log 9 (x+a)) (-a+1, 0)

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25
Q

Discriminant

A

b^2-4ac >0 2
b^2-4ac = 0 1

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26
Q

Proof for sum of arithmetic series

A
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27
Q

Proof for sum of a geometric series

A
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28
Q

Proof for sum to infinity

A
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29
Q

Sigma notation proof

A
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30
Q

Sin (2A)

A

2SinACosA

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31
Q

Cos(2A)

A

Cos^2A-sin^2A
2cos^2A-1
1-2sin^2A

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32
Q

Tan2A

A

2tanA/1-tan^2A

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33
Q

Sin4X

A

2Sin2Acos2A

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34
Q

1-cos^4A

A

(1+cos^2A)(1-cos^2A)

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35
Q

E = km + b

A

K is the increase in E when m increase by 1
b is E at the beginning or the fixed charged for E

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36
Q

Y = f(x) + a

A

X, y+a

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37
Q

Y = f(x + a)

A

X-a, y

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38
Q

Y = af(x)

A

X, ay

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39
Q

Y = f(ax)

A

X/a , y

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40
Q

Y = -f(x)

A

X, -y

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41
Q

Y = f( - x)

A

-x, y

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42
Q

Gradient for parallel and perpendicular

A

Parallel = same gradient
Perpendicular = neg reciprocal

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43
Q

Y=kx

A

If x increases by 1 unit, y increases by k units

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44
Q

Ncr formula

A

N!/r! (N-r)!

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45
Q

Cosine rule

A

A^2 = b^2 + c^2 -2bc cos A

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46
Q

Area of triangle

A

1/2 ab sinC

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47
Q

Y = sinx

A
  • repeat’s every 360 and crosses the x axis at (-180, 0, 180, 360)
  • maximum of 1 and minimum value of -1
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48
Q

Y= cosx

A
  • repeat’s every 360 and crosses the x axis at (-90, 90, 270, 450)
  • maximum of 1 and minimum value of -1
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49
Q

Y=tanx

A
  • repeat’s every 180 and crosses the x axis at (-180, 0, 180, 360)
  • has no maximum or minimum
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50
Q

Unit of a vector

A
  • size of vector is 1
  • a (2
    3). |a| = √3

Unit = (2/√3)
3/√3）

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51
Q

Collinear

A

Lie of the same line

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52
Q

Find the gradient of the tangent to the curve

A
  • Find f’(x) = dy/dx
  • Sub the given point into f’(x) = gradient
  • Form equation
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53
Q

Find the gradient of the normal to the curve

A
  • Find f’(x) = dy/dx
  • Sub the given point into f’(x) = gradient
  • negative reciprocal of gradient
  • Form equation
54
Q

Increasing function

A

If f’(x) ≥ 0

55
Q

Decreasing function

A

If f’(x) ≤ 0

56
Q

Interval of which a function is decreasing

A

F(x) = x^3+3x^2-9x
F’(x) = 3x^2+6x-9 ≤ 0
3(x+3)(x-1)≤0
-3≤x≤1

57
Q

Find a stationary point

A
  • Find dy/dx and set to 0 to find an x value
  • sub the x value into the original equation f(x) to find y value
58
Q

Finding out whether a stationary point is a local minimum or maximum or point of inflection method 1

A

Stationary point (a,b)

F’’(a) < 0 max
F’’(a) = 0
F’’(a) > 0

59
Q

Finding out whether a stationary point is a local minimum or maximum or point of inflection method 2

A

Stationary point (a,b)

Consider points just below and above x and sub into f’(x)

60
Q

Finding the constant of integration (c)

A
  • integrate the function
  • sub the point on curve into integrated fuction
  • solve to find c
61
Q

Sketching gradient functions

A

y=f(X) y=f’(X)
Maximum or minimum cuts the a xis
Point of inflection touches the x axis
Positive gradient above the x axis
Negative gradient below the x axis
Vertical asymptote same
Horizontal asymptote y=0

62
Q

Definite integral (area)

A
63
Q

Modulus function graphs

A

Y = |f(x)|
Reflected on x axis

64
Q

Arc length

A

R theta

65
Q

Area of sector

A

1/2 r^2 theta

66
Q

Natural numbers

A

N
only the positive integers i.e. 1, 2, 3, 4,5,6, ………. excluding zero, fractions, decimals and negative numbers

67
Q

Z

A

Integers
A whole number

-2, -1, 0, 1, 2, 3

68
Q

Q/R

A

Rational numbers

Anything that can be written asa fraction of 2 integers

69
Q

R

A

Real numbers

Anything

70
Q

Null set

A

Nothing in set, empty

71
Q

Element of a set

A

E
Included in set

72
Q

Not an element of a set

A

E with strike

73
Q

P

A

Irrational numbers
Cannot be written as a fraction of integers
Pi, sqrt (5)

74
Q

Z+

A

Positive integers

75
Q

State one limitation of the model

A

It is unlikely that the car’s value with decrease with age
The lorry won’t drive at the same pace throughout the whole journey

76
Q

Tangent

A

Only touches once

77
Q

Congruent

A

Same length

78
Q

Scalene

A

Has 3 different lengths

79
Q

What’s the shortest distance between two parallel lines

A

The perpendicular distance

80
Q

Sin(x)cos(x)

A

0.5sin(2x)

81
Q

Explain why the function has no inverse?

A

The function is many to one

82
Q

8sinxcosx

A

4(2sinxcosx)
= 4sin2x

83
Q

How to find gradient of curve?

A

Differentiate and sub in point

84
Q

Find the exact range of values of x for which the curve is increasing.

A

Differentiate and set > 0

85
Q

Explain relationship between gradients of points on curve

A

As h —> 0, 12+ 3h —> the gradient of the chord tends to the gradient of the tangent to the curve

86
Q

Show that the curve has no stationary points

A

Differentiate
- use the Discriminant to show ≠ 0 hence the curve has no turning points

87
Q

Verify the curve has a stationary point when x=4

A

Differentiate and sub in 4 which should = 0

88
Q

Determine the nature of this stationary point

A

Differentiate twice and sub in x >0 is a minimum

89
Q

How to find turning point of cubic.

A

Differentiate and factorise the quadratics

90
Q

When to add c for integration?

A

First of all remember that you only need to include the +c if you are integrating without limits (ie you don’t have the two numbers at the top and bottom of the integration symbol). We differentiated A by differentiating each small part of it separately and adding them together at the end.

91
Q

Find the minimum perimeter of the pool

A

Differentiate and set to 0 to find value for x and sub into equation

92
Q

Increasing Interest rates

A

Starting value is 20,000
8% interest increase
(20,000x1,08) year 2
(20,000x1.08^2)

Value of r = 1,08

93
Q

Decrease in value of car

A

Starting value is 20,000
8% decrease
(20,000x0.08) year 2
(20,000x0.08^2)

Value of r = 0.08

94
Q

Condition for an infinite geometric series with common ratio r to be convergent

A

|r| < 1

95
Q

Surface are of cyclinder

A

H= height
R= radius

2πrh+2πr2

96
Q

Volume of sphere

A

4/3πr^3

97
Q

Volume of cyclinder

A

πr2h

98
Q

Area of rhombus

A

diagonal lines times together and divide by 2

98
Q

Area of parrellogram

A

base x perpendicular height

98
Q

Concave function

A

If the second derivative is ≤ 0
f’’(x) ≤ 0

99
Q

Convex function

A

If the second derivative is ≥ 0
F’’(x) ≥ 0

100
Q

Two odd numbers

A

2n+1, 2m+1

101
Q

Difference of two squares

A

A^+b^+2ab
Cos^2+sin^2+2coxsinx (cosx+sinx)

102
Q

Increase by percentage (sequence)

A

2100 is starting, increase by 1.2%
Find from 2017-2030 (include the 2017)
1.2/100 +1 = 1.012
2100((1.012)^14-1/1.012-1

103
Q

Why can’t you use turning point, stationary point on curve (newton raphson)

A

The gradient is 0 = tangent never touches the x axis which is required by x axis

104
Q

differentiate 2xy with respect to x

A

2y + 2x(dy/dx)

105
Q

Rate of change of depth

A

Dh/dt

106
Q

Vertical line gradient

A

M = infinity
Dy/dx = infinity
Dx/dy = 0

107
Q

Horizontal line

A

Dy/dx = 0

108
Q

Differentiate cos, sin from first principles

A
109
Q

Rate of flow of water

A

Dv/dt

110
Q

Parametric equation domain and range

A

The domain of f(x) is the range of x
The range of 𝑓(𝑥) is the range of y

111
Q

Determine whether or not this iteration formula can be used to find an approximation for a, justifying your answer

A
  • annotate the graph by drawing cobweb diagram
  • “the iteration formula can be used to find an approximation for a because the cobweb spirals inwards for the cobweb diagram”
112
Q

Prove by counter example: prove for all positive real values of x….

A

Let x = 1/8
Let x = 6

113
Q

Rocket is described with equation. Show that the rocket returns to the ground between 19.3 and 19.4 seconds after launch

A

Sub 19.3 and 19.4 show there is a change of sign

114
Q

Figure 1 is a graph of the price of a stock during a 12-hour trading window. The equation of the curve is given above. Show that the price reaches a local maximum in the interval .

A

Differentiate equation and then sub

115
Q

Explain why (for certain equations) can the Newton Raphson method cannot be used?

A

Accept any reasons why the Newton-Raphson method cannot be used with x1 = 0 which refer or allude to either the stationary point or the
tangent. E.g.
- There is a stationary point at x = 0
- Tangent to the curve (or y = 2x3  x2 =) would not meet the x-axis

116
Q

For binomial limits (modulus) in fractions, which value do you use?

A

The smaller value

117
Q

P(AnB) for independent

A

P(A) x P(B)

118
Q

P(A|B)

A

P(AnB)/P(B8

119
Q

Non mutually exclusive/independent P(AUB)

A

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

120
Q

Mutually exclusive P(AuB)

A

P(A) + P(B)

121
Q

Mutually exclusive P(AnB)

A

P(AnB) = 0

122
Q

Which value is more accurate when subbing in x for a binomial expansion?

A

The smaller the x value, the more accurate

123
Q

Explain how the trapezium rule could be used to obtain a more accurate estimate for the area of S.

A

Increase the number of strips

124
Q

Explain why, for this question, the Newton-Raphson method cannot be used with x1 = 0

A

There is a stationary point at x=0
- the tangent to the curve would not meet the x axis

125
Q

Find turning point

A

Differentiate and set to 0

126
Q

Find the minimum perimeter of the pool, giving your answer to 3 significant figures.

A

Differentiate and set to 0 then sub back into the equation

127
Q

Surface area of a shape

A

Area of each face and add together

128
Q

Surface area of a cylinder

A

2πrh+2πr^2

129
Q

Volume of sphere

A

4/3πr^3

Maths (5 decks)

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