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A - Level Maths > Pure > Flashcards

Pure Flashcards

1
Q

What is the domain

A

The domain of a function is the set of possible inputs

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2
Q

What is the range of a function

A

A set of possible outputs

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3
Q

What are roots of a functions

A

The values of x oh which f(x) = 0

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4
Q

What does
X E r(with an extra line)
Mean in functions

A

X - input x…
E- is a member of …
R- the set of real numbers…

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5
Q

What does lN mean

A

Natural numbers - aLl positive intergers

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6
Q

What does z(with a lien) mean

A

All intergers

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7
Q

What is an asymptote

A

A line which approaches the graph but never reaches

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8
Q

What is the order of polynomials

A

0 - constant
1 - linear
2 - quadratic
3 - cubic
4 - quartic
5 - Quintic

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9
Q

The number of turns is ……….. than the order

A

One less

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10
Q

What does x ——-> infinity shape

A

As x tends to infinity

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11
Q

If a>0
And the Order Is odd described the graph

A

It goes uphill from left to right

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12
Q

If a>0
And the Order Is even described the graph

A

The tale goes upward

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13
Q

Y = f(x) + a
Describe the transformation mathematically

A

Translation
( 0)
( a)

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14
Q

Y = f(x) -a

Describe the transformation mathematically

A

Translation
(0)
(-a)

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15
Q

Y = f(x+a)

Describe the transformation mathematically

A

Translation
(-a)
(0)

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16
Q

Y = f(x-a)
Describe the transformation mathematically

A

Translation
(a)
(0)

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17
Q

Y = -f(x)

Describe the transformation mathematically

A

Reflection in x axis

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18
Q

Y = f(-x)

Describe the transformation mathematically

A

Reflection in your axis

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19
Q

Y = af(x)

Describe the transformation mathematically

A

Stretch parallel to the y axis with a scale factor of a

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20
Q

Y = f(ax)

Describe the transformation mathematically

A

Stretch parallel to x a us with scale factor 1/a

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21
Q

Y = f(x) +a
Describe the transformation in words

A

Moves graph a. Units up

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22
Q

Y = f(x) -a

Describe the transformation in words

A

Moves graph a units down

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23
Q

Y= f (x+a)

Describe the transformation in words

A

Moves graph a units to the left

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24
Q

Y = f (x-a)

Describe the transformation in words

A

Moves graph a units to the right

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25
Q

Y = -f(x)

Describe the transformation in words

A

Flips graph over
Makes all positive y coordinates negative and vice versa

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26
Q

Y = f(-x)

Describe the transformation in words

A

Makes all positive x coordinates negative and vice versa

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27
Q

Y = af(x)

Describe the transformation in words

A

Makes graph taller and skinnier
All y values are multiplied by a with x values unchanged

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28
Q

Y = f(ax)

Describe the transformation in words

A

Squashes graph in if a is bigger than 1; stretches graph out if a is between 0 and 1. All x values are divided by a, with y values unchanged

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29
Q

What other way could you write the equation m= y-yi/ x- Xl,

A

Y-yi =m(x- xi)

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30
Q

The gradient of parrauel lines are?

A

Equal

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31
Q

If two lines are perpendicular the the gradient of one is the ——— of the other

A

Negative reciprocal

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32
Q

What is a vector

A

Magnitude and direction

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33
Q

What is magnitude of a vector

A

The length of the line

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34
Q

What do u get if you add 2 vectors

A

A resultant vector

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35
Q

Multiplying a vector by a non-zero scalar produced a
…………. vector

A

Parallel

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36
Q

How do you show 2 vectors are parallel

A

Show there scalar multiples of each other

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37
Q

What is a position vector

A

Position of a point in relation to the origin

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38
Q

What is the position vector of A

A

— >
OA

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39
Q

What is a unit vector

A

Vector with magnitude one

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40
Q

What are the standard unit vectors

A

i
j

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41
Q

i is rhe direction of the …. Axis (vectors)

A

x

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42
Q

j is rhe direction of the …. Axis (vectors)

A

Y

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43
Q

Xi + Yj =
As a column vector

A

(X)
(Y)

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44
Q

Vector xi + yj has a magnitude
And angle ……. With horizontal

A

Square root x2 +y2
Tan-1 (y/x)

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45
Q

How is magnitude written

A

|a|

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46
Q

How is a unit vector written

A

a = a/ |a|

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47
Q

What are the 3 equations of a line

A

Y-Y1 = m (x - x1)

Y =mx +c

ax +by +c = 0

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48
Q

What are similar about parallel lines

A

They have equal gradients

49
Q

What are the gradients like in perpendicular lines

A

The negative reciprocal

50
Q

How do you turn X^2 + y^2 +2fx + 2gy + c =0 into a familiar form of a circle

A

Complete the square by collecting like terms first

51
Q

What is the equation of a circle

A

(X-a)2 + (y-b)2 = r2

52
Q

Whatare the 3 circle theorems you need to know

A

1) the angle in a semi circle is a right angle
2) the perpendicular line form the centre to a chord bisects the chord
3) a radius and tangent to the same point will meet at right angles

53
Q

How to find the equation of a tangent

A

It will be perpendicular to radius so find gradient of radius and find perpendicular gradient
Then substitute in the point on the tangent to complete

54
Q

How do you find the centre of a circle with a triangle in

A

The triangle sides are chords so there perpendicular bisectros meet in the centre

So find perpendicular bisectors id 2 sides
Find rhe midpoint + gradient to find equation
Then find centre by simultaneous equation

55
Q

What is disproof by counter example

A

While to prove a statement is true we need to prove every possible case. We only need to disprove one example to disprove a statement
(Known as a counter example)

56
Q

What is proof by exhaustion

A

This is breaking down the state but into all possible smaller cases where we prove each individual case
( sometimes known as case analysis )

57
Q

What is an identity

A

An equation that is true cal all the values of the variable

58
Q

What is a conjecture

A

A mathematical statement that has yet to be proven

59
Q

What is a theorem

A

A mathematical statement that has been proven

60
Q

What should a proof show

A

All assumptions
A sequenced list of steps that logically follow and must cover all possible cases
+ make a concluding statement

61
Q

What is proof by deduction

A

Start from known facts and reach the desired conclusions via deductive steps

62
Q

What are the 3 types of proof

A

Proof by deduction
Proof by exhaustion
Disproof by counter example

63
Q

What is rhe dividend

A

The thing your dividing

64
Q

What is the degree

A

The highest power of x in the polynomial

65
Q

What is the divisor

A

This is what your dividing by

66
Q

What is a quotient

A

What you get when you divide by the divisor ( not inc the remainder

67
Q

What is the remainder

A

What’s left over

68
Q

……. Degrees = …….. radians

A

180
Pi

69
Q

How do you turn radians to degrees

A

Divide by pi multiply by 180

70
Q

How do you turn degrees to radians

A

Divide by 180
Multiply by pi

71
Q

What is a unit circle

A

A circle with radius 1, centre on the origin

72
Q

What can any point on a unit circle be defined as

A

( cos theater, sin theater )

73
Q

Sector area

A

A = 1/2 r^2 theater
Or
Theater / 360 x Pi r ^2

74
Q

Arc lenght equation

A

S = r theater
Or
2 pi r ^2 x theater/360

75
Q

Sine rule equation

A

a. b. c
———. =. —— — =. — ——
SinA. SinB. SinC

76
Q

Cosine equation

A

a2 = b2 + c2 - 2bc cos A

77
Q

Rearranged cosine equation for angle

A

Cos A = b2 + C2 -a2
— - - ——-
2bc

78
Q

What are the trig identities

A

Tan x = sinx / cos x

Sin^2x + cos2^x =1

79
Q

When f(x) is decreases
Is f’(x) positive or negative

A

Negative

80
Q

When f(x) is increasing
Is f’(x) positive or negative

A

Positive

81
Q

When the gradient dy/dx of a graph is positive the value of y is ……

A

Increasing

82
Q

What is a stationary point

A

Where the gradient is 0

83
Q

What is the local maximum

A

The gradient is posotive to the left, zero at the point , and negative to the right

84
Q

What is the local minimum

A

The gradient is negative to the left, zero at the point, and positive to the right

85
Q

How can you distinguish between local max and local min

A

You can test values of the derivative either side of the stationary point, to see where gradient is positive or negative

86
Q

What is the point of infection

A

Where the curve moves from convex to concave (or vide versa)

87
Q

A point of infection which is a stationary point Is knwon as what

A

Saddle point

88
Q

At a point of inflection, is the gradient postives or negative just before or after the stationary point

A

Just before and just after positive and positive
or
negative and negative

89
Q

What is the gradient at a station point

A

0

90
Q

If a function is increasing the gradient is …

A

Positive

91
Q

If a function is decreasing the gradient is …

A

Negative

92
Q

What does
d
—
dx
mean?

A

Derivative in respect to x

93
Q

f( x) derives to

A

f’(x)

94
Q

Y dervives to

A

dy
—
dx

95
Q

A constant always differentiates to

A

0

96
Q

What does differntatiating tell you

A

The gradients of a curve at any point

97
Q

What is differentiate first principles

A

Lim = f(x+h)-f(x)
h—>0. (.—————)
h

98
Q

How do you use differntiate first principles

A

Find, simplify, remove h from denominator
Find limit of the expression as h tends to 0, by h=0

99
Q

How do you find the nature if a turning point

A

Differentiate again
f’’(x) > 0 it’s minimum
f’’(x) <0 its maximum

100
Q

what is the second derivative of y

A

d2y
____
dx2

101
Q

What is the second derivative of f(x)

A

f’’(x)

102
Q

What do you get if you integrate dy/dx

A

y + c

103
Q

How do you integrate

A

increase powers by one
Then divide constant by power
Add c

104
Q

nCr = …..

A

n!
———
r!(n-r)!

105
Q

Does the first term in a bingo expansion descend or ascend

A

Descend
The second term ascends

106
Q

What is an increasing sequence

A

Each term is greater then the one before

107
Q

What is a decreasing sequence

A

Each term is less then the other one

108
Q

What is an arithmetic sequence

A

The differnce between one therm and the next is always the same

109
Q

What is a geometric sequence

A

The ratio of one term to the next is always the same

110
Q

What is a periodic sequence

A

Repeats itself at regular intervals. The number of terms before the sequence repeats is called the period

111
Q

What is a series

A

The sun of the terms of a sequence

112
Q

What 2 ways can sequences been defined

A

Inductively or deductively

113
Q

What is a deductive definition

A

Gives a direct formula for the Kth term of the sequence in terms of K. The terms of the sequence can be found by substitution the numbers 1,2,3,4 for K

114
Q

What is an inductive definitions

A

Tells you have to find a term in a sequence from the previous term - the definition must also inculde the value of the first term of the sequence, you can find the second term form the first term and the third term form the second term …

115
Q

What is the formula for ak, the kth term of the sequence

A

ak - a + (k-1)d

116
Q

Ensure you can dervie arithmetic formula for Sn

A

.

117
Q

What is an arithmetic series

A

The sum of the terms of an arithmetic sequence

118
Q

What is the modulus of a number

A

.the non negative numerical value

A - Level Maths (4 decks)

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