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Maths A-Level > Pure > Flashcards

Pure Flashcards

1
Q
A
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4
Q

∆

A
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5
Q

∆ > 0

A

2 real distinct roots

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6
Q

∆ = 0

A

1 repeated root

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7
Q

∆ < 0

A

no real roots

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8
Q

the natural numbers

A
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9
Q

the integers

A
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10
Q

the rationals

A
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11
Q

the real numbers

A
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12
Q

A is a subset of B

A

A ⊂ B

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13
Q

0 < x ≤ 10 in interval notation

A

(0,10]

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14
Q

The set of integers between 1 and 100 inclusive

A

{1,2,3,…,100}

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15
Q

Is a member of

A

∈

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16
Q

The number of members in set A is 5

A

n(A) = 5

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17
Q

Universal set

A

ξ

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18
Q

Complement of A

A

A’

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19
Q

Union of A and B

A

A ∪ B

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20
Q

Empty set

A

Ø

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21
Q

A is not a subset of B

A

A ∉ B

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22
Q

x is a real number and is less than 10

A

{x ∣ x∈N, x<10}

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23
Q

Modulus function

A

∣x∣

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24
Q

one-one

A

if every y value corresponds to only one x value

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25
Q

many-one

A

if there is at least one y value that comes from more than one x value

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26
Q

domain

A

the set of allowed input values to a function

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27
Q

range

A

the set of all possible outputs of a function

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28
Q

one-many

A

if there is at least one x value that gives more than one y value

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29
Q

a mapping is a function if…

A

every input value maps to a single output value

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30
Q

how can you test whether a mapping is a function?

A

use the vertical line test

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31
Q

mapping

A

takes numbers from a given set and assigns each of them one or more output values

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32
Q

image

A

the output of a given input

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33
Q

y = f(x) + c

A

translation c units up

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34
Q

y = f(x+d)

A

Translation d units to the left

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35
Q

y = af(x)

A

Vertical stretch by scale factor a

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36
Q

y = f(ax)

A

Horizatal stretch by 1/a

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37
Q

y = -f(x)

A

reflection in the x-axis

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38
Q

y = f(-x)

A

reflection in the y-axis

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39
Q

order of operation for vertical transformations (e.g. y = af(x) +c)

A

normal order of operations

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40
Q

order of operations for horizontal transformations (e.g. y = f(ax + c))

A

reverse of normal order of operations

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41
Q

order of operations for one horizontal and one vertical transformations (e.g. y = f(ax) + b)

A

doesn’t matter

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42
Q

equation of a straight line

A

y = mx + c

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43
Q

gradient of normal

A
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44
Q
A

(a) (a,b)
(b) c

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45
Q

If the hypotenuse of a triangle within a circle is the diameter…

A

It is a right angled triangle

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46
Q

the radius of a circle at a given point on its circumference is…

A

perpendicular to the tangent to the circle at that point

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47
Q

increasing sequence

A

a sequence where each term is larger than the previous one

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48
Q

decreasing sequence

A

a sequence where each term is smaller than the previous one

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49
Q

periodic sequence

A

one where the terms start repeating after a while

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50
Q

finite sequence

A

has a finite number of terms

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51
Q

infinite sequence

A

continues forever (infinite number of terms)

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52
Q

term-to-term rule

A

e.g.

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53
Q

position-to-term rule

A

formula for the nth term

e.g.

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54
Q

converge

A

tend to a certain value as n increases

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55
Q

diverge

A

increase/decrease without limit

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56
Q

series

A

the sum of a sequence up to a certain point (S(n))

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57
Q
A
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58
Q

Write S(n) in sigma notation

A
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59
Q

arithmetic sequences

A

sequences in which there is a common difference between each term (e.g. +4)

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60
Q

the nth term of an arithmetic sequence =

A
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61
Q

sum of the first n terms for an arithmetic sequence (a & d) =

A
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62
Q

geometric sequence

A

a sequence in which there is a common ration between each term (e.g. x2)

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63
Q

the nth term of a geometric sequence =

A
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64
Q

sequence notation - what does a mean?

A

the first term

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65
Q

sequence notation - what does d mean?

A

the common difference

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66
Q

sum of the first n terms for a geometric sequence =

A
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67
Q

sequence notation - what does r mean?

A

common ratio

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68
Q

sum to infinity of a geometric series

A
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69
Q

under what conditions does the sum to infinity condition apply?

A

|r| < 1 (series converges)

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70
Q

if |r| > 1 …

A

the geometric series diverges

71
Q

sequence notation - what does L mean?

A

the last term in a sequence

72
Q

sum of the first n terms for an arithmetic sequence (a & L) =

A
73
Q
A
74
Q
A
75
Q
A
76
Q
A
77
Q
A

0 (for any base a>0)

78
Q
A

1 (for any base a>0)

79
Q

Direction of a vector

A

The angle the vector makes with the horizontal

80
Q

Displacement vector

A

The vector representing the translation from one point to another

81
Q

Direct proportion

A

A relationship between two variables in which their quotient is constant

82
Q

Congruent expressions

A

Expressions connected by an identity symbol (they are equal for all values of the variable)

83
Q

Coefficient

A

A constant in front of a variable

84
Q

Collinear

A

Points that lie along the same straight line

85
Q

Chord

A

The line segment between two points on a curve

86
Q

Exponential decay

A

A relationship of the form y=e-x

87
Q

Exponential equation

A

An equation with the ‘unknown’ variable in the power

88
Q

Exponential growth

A

A relationship of the form y=ex, where a>1

89
Q

Factorial

A

The product of integers from 1 to n, denoted by n!

90
Q

Identity

A

A relation which is true for all values of the unknown

91
Q

Interval notation

A

A form of notation to represent all numbers in a range - uses () if point is not included and [] if it is

92
Q

Inverse proportion

A

A relationship between two variables in which their product is constant

93
Q

Lead coefficient

A

The coefficient of the highest degree term in a polynomial

94
Q
A

a < x < b

95
Q
A

‘P implies Q’ or ‘If P is true then Q is true’

96
Q
A

‘P is equivalent to Q’ or ‘Q is true is and only if P is true’

97
Q

What is the degree of a polynomial?

A

The highest power of the unknown (x) occuring in the function

98
Q

For a quadratic function how many…

(a) x-intercepts
(b) turning points

A

(a) 0-2
(b) 1

99
Q

For a cubic function how many…

(a) x-intercepts
(b) turning points

A

(a) 1-3
(b) 0 or 2

100
Q

For a quartic function how many…

(a) x-intercepts
(b) turning points

A

(a) 0-4
(b) 1 or 3

101
Q
A
102
Q
A
103
Q

Direct proportion

A

Ratio of the two quantities is constant

104
Q

What is the relationship between y and x2 if they are directly proportional?

A
105
Q

What is the relationship between y and x2 if they are inversely proportional?

A
106
Q

What are the 3 key circle angle rules?

A

1) The angle in a semi circle is a right angle
2) A tangeant to the circle is perprendicular to the radius at the point of contact
3) The radios perpendicular to the chord bisects the chord

107
Q

What is the distance between the points (x1, y1) and (x2, y2)?

A
108
Q

How can you tell whether two circles intersect, are disjoint ot tangeant to each other?

A

Compare the difference of the radii of the circles to the distancce betweeen their centres

109
Q

What is a=bc in log form?

A

c=logba

110
Q

Taking a logorithm of which numbers produce non-real numbers?

A

A negative number or zero

111
Q

What is log10x written as?

A

log x

112
Q

What is logex written as?

A

ln x

113
Q

loga(ax)

A

x

114
Q
A

x

115
Q

What is true of all graphs of the form y = ax ?

A

1) The y-intercept is (0,1)
2) The graph of the function lies entirely above the x-axis
3) The x-axis is an asymptote

116
Q

What is the gradient of ex ?

A

ex

117
Q

What is the gradiant of e<em>kx</em> ?

A

k ekx

118
Q

Graph of y = ln(x)

A
119
Q

What is true of the graph y = ln(x) ?

A

1) Passes through the point (0,1)
2) The y-axis is a vertical asymptote

120
Q

What is true of the function of the form y = Aekt ?

A

1) The initial value (when t=0) is A
2) The rate of change is ky

121
Q

If y = kbx then…

A

log y = log k + x log b

122
Q

For equation log y = log k + x log b, the graph of log y against x is a straight line with what

(a) gradient?
(b) y-intercept?

A

(a) log b
(b) log k

123
Q

2 key angle identities

A

1) tan x = sin x / cos x
2) sin2 x + cos2 x = 1

124
Q

Sine rule

A
125
Q

Cosine rule

A
126
Q

What do you have to look out for when using the sine rule?

A

There may be two possible answers (A and 180-A)

127
Q

Area of a triangle

A
128
Q

What is true if vectors a and b are parallel?

A

b = ta

129
Q

Unit vector

A

A vector with magnitude 1

130
Q

Position vector

A

Gives the vector from the origin to that point (its coordinates)

131
Q

How can vectors be used to prove geometric shapes?

A

1) If a shape is a parallelogram then the vectors corresponding to the opposite sides are equal
2) If a shape is a rhombus then the vectors corresponding to all four sides have equal magnitudes

132
Q

The midpoint of a line joining points with position vectors a and b had position vector…

A

1/2 (a + b)

133
Q

How can the compisite function applying g to x and then f be written?

A
134
Q

How is the inverse of a function written?

A
135
Q

How do you write the derivative of a function?

A

f’(x)

136
Q

How does the graph of y=f-1(x) relate to y=f(x)?

A

y=f-1(x) is a reflection of the graph y=f-(x) in the line y=x

137
Q

What can you say about the range domain of f-1(x) and f(x)?

A

1) The domaine of f-1(x) is the same as the range of f(x)
2) The range of f-1(x) is the same as the domain of f(x)

138
Q

What order are the transformations done for the function y = p f(x) + c ?

A

Stretch is performed before the translation

139
Q

What order are the transformations done for the function y = f(qx + d) ?

A

The translation is performed before the stretch

140
Q

Rational function

A

A fraction where both the denominator and numerator are polynomials

141
Q

What is the relationship between degrees and radians?

A

180° = π radians

142
Q
A
143
Q
A
144
Q

Converting from degrees to radians

A

Divide by 180º and multiply by π

145
Q

Converting from radians to degrees

A

Divide by π and multiply by 180º

146
Q

For the function y = a sin (bx) and y = a cos (bx) what is the

(a) period
(b) amplitude

in radians?

A

(a) 2π/|b|
(b) a

147
Q

For the function y = a sin (b(x+c)) + d and y = a cos (b(x+c)) what is the

(a) period
(b) amplitude
(c) central value
(d) minimum value
(e) maximum value

in radians?

A

(a) 2π/|b|
(b) |a|
(c) d
(d) d - |a|
(e) d + |a|

148
Q

Arc length in radians

A
149
Q

Arc length in degrees

A
150
Q

Area of a sector in degrees

A
151
Q

Area of a sector in radians

A
152
Q

Segment on a diagram

A
153
Q

Sector on a diagram

A
154
Q

Double angle identity for sin

A
155
Q

Double angle identity for cos

A
156
Q

Double angle identity for tan

A
157
Q

Compound angle identity for sin

A
158
Q

Compound angle identity for cos

A
159
Q

Compound angle identity for tan

A
160
Q

Harmonic form

A
161
Q

sec x

A

1/cos x

162
Q

cosec x

A

1/sin x

163
Q

cot x

A

1/tan x

164
Q

Identities for reciprocal trigonometric functions

A

Differentiation of sinx, cosx and tanx

165
Q

Differentiation of ln x

A
166
Q

Differentiation of trigonometric functions

A

What is the product rule?

167
Q

Differentiate akx

A
168
Q

Differentiate cos(kx)

A
169
Q

Differentiate ekx

A
170
Q

Differentiate sin(kx)

A
171
Q

Differentiate tan(kx)

A
172
Q

Implicit differentiation

A
173
Q

Which two special cases of integration by substitution should be remembered?

A
174
Q

Integration by parts

A

Maths A-Level (3 decks)

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