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A Level Further maths > mock\ > Flashcards

mock\ Flashcards

1
Q

1.

what is an algorithm?

A
  • a finite sequence of operations for carrying out a procedure/solving problems
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2
Q

what is a heuristic algorithm?

A

a method which finds the solution efficenlty , with no gurantee that it is optimal solution

eg: packing algorithms, integer LP

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3
Q

what is optimisation?

A

optimal= “best”
optimisation= making it best
* finding the best availible values

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4
Q

when is a LP in standard form?

A
  • Objective function to be maximised
  • with constraints <=
  • non negative variables
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5
Q

what is augmented/ slack form?

A

non-negative slack variables convert ineualities to equalities

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6
Q

what are basic variables?

A

variables which have been cleared out and only have one non zero column in it

hold and actual value

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7
Q

what are non-basic variables?

A

variables which have multiple values in thier column and so are given a 0 value

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8
Q

when would a LP require integer solutions?

A

when the variable is discrete
* should be mentioned in working out

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9
Q

What does this sigma notation mean?

A
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10
Q

Sigma notation

A
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11
Q

Sigma notation

A
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12
Q

Sigma notation

A
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13
Q

Express the sum using Sigma notation

A
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14
Q

The standard formula for the sum to n terms of r

A
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15
Q

Find the sum

A
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16
Q
A
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17
Q

The standard formula for the sum to n terms of r squared

A
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18
Q
A
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19
Q

Find a general expression for

A
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20
Q
A
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21
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22
Q
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23
Q

Method of differences

A
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24
Q
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25
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26
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27
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28
Q

What are the general steps for proof by induction?

A
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29
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30
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31
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32
Q

Proof by induction factorial

A
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33
Q

Proof by induction product

A
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34
Q

What is the order of the following matrices ?

A

Number of rows by numbers of columns
y by x

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35
Q

What does it mean for matrices to be “non conformable”?

A

This means that they matrices do not have the exact same order. For example

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36
Q

What does associative mean?

A

It means that despite the order/ grouping of numbers, the answers will be the same.

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37
Q

What does it mean for something to be commutative ?

A
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38
Q

What are the general steps to carrying out matrices multiplication?

A
  1. write out the order of both matrices
  2. see if compatible, if they are write order of answer matrix
  3. start with first element (1x1) first row by first colum
  4. multiply each pair of numbers and add up the answers
  5. then go onto next element (1x2) first row by second column
  6. multiply each pair and add up numbers
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39
Q

Multiply

A
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40
Q

Multiply (find FE)

A
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41
Q

Is matrix muliplication commutative or associative?

A
  • not commutative
    as order within which multiplied does matter
  • is associative
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42
Q

what is the original point/shape and the transformed point/shape reffered to ?

A
  • orginal= object
  • transformed= image
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43
Q

what is a transformation?

A

the mapping of an object onto a image

44
Q

how do you find the transformation representated by a given matrix?

A
  1. split up the identity matrix into points (unit vectors)
  2. see how they have been transformed by ploting the points on a graph
45
Q

Describe the transformation of the matrix

A
46
Q
A
47
Q

What is the transformation matrix for any angle theta anti-clockwise. ?

A
48
Q

What is the matrix transformation for any clockwise rotation by angle theta?

A
49
Q

Describe the transformation by the matrix

A
50
Q

Describe the transformation by the matrix

A
51
Q

What are shears?

A

The base stays rooted and the rest shifts and becomes parallel. The area of original shape is the same to the shear.

52
Q

What are the 2 different types of shears?

A
53
Q

Successive transformations

A
54
Q

3D matrices

A
55
Q

Describe the Transformation under matrix

A
56
Q

Describe the Transformation under matrix

A
57
Q

Describe the transformation

A
58
Q

What are invariant points?

A

Points that do not change after a transformation.
The origin will always be an invariant point

59
Q

What are invariant lines?

A

A line of points that MAY be invariant or points that remain upon that line after a transformation

60
Q
A
61
Q

More proof by induction matrices

A
62
Q
A
63
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A
64
Q

What are the components of an algorithm?

A
  • finite number of steps
  • unambigious steps
  • deterministic
65
Q

What is complexity ?

A
  • Size in relation to run time (efficiency )
  • measure of which one is faster
66
Q

Perform a quick sort algorithm to sort it into ascending order

A

values less than or equal to go to the left of pivot

67
Q

what is
* first fit
* first fit decreasing
* full bin

A
  • FF: putting the numbers using given order in question
  • FFD: ordering the list into descending order - biggest first
  • FB: using brain for optimal solution
68
Q

what does heuristic mean?

A
  • a method that finds the solution efficently, but with no gurante that this solution is optimal
  • important when classical methods fail, only way to find is exhauting all solutions
69
Q

how can the efficency of different packing methods be tested?

A

counting the number of comparisions needed in worst case

70
Q

what is the complexit of the first fit and first fit decreasing algorithm , in the worst case?

A

o (n sqaured)

71
Q

what is the complexity of the quick sort algorithm , in the worst case?

A

O (n squared)

72
Q

What is big O notation?

A
73
Q
A
74
Q

Use quick sort to sort the following numbers in descending order

A
75
Q

Use quick sort to sort the following numbers in descending order

A
76
Q

What is i?

A

The square root of negative 1

77
Q

What is an do the following symbols mean ?

A
78
Q

What is an imaginary number ?

A

Any multiple of i

79
Q

What is a complex number ?

A

An imaginary number and a normal number

80
Q

How is the equality different for complex numbers?

A
81
Q

Questions

A
82
Q

Questions

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83
Q

Question

A
84
Q
A
85
Q
A
86
Q

What is a conjugate ?

A

A change in the sign in the middle of two terms

87
Q
A
88
Q
A
89
Q
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90
Q
A
91
Q
A
92
Q

What do you know about the roots of a quadratic equation that are non- real complex numbers?

A

The roots are conjugate.
If f(z)=0
Then f(z*)=0

93
Q
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94
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95
Q
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96
Q
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97
Q
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98
Q
A
99
Q

What is the Argand diagram?

A
100
Q

Plot the points

A
101
Q

How would two conjugate pairs be represented geometrically? (On the Angrand diagram)

A

Reflection on the x axis

102
Q
A
103
Q
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104
Q
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105
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106
Q
A

A Level Further maths (10 decks)

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