CAP 1

Question 1

What is a matrix?

Answer 1

A matrix is an array of elements set out in a pair of brackets.

Question 2

How many rows and columns does a MxN matrix have?

Answer 2

M rows and N columns

Question 3

What is a square matrix

Answer 3

A matrix that has the same number of rows as columns

Question 4

What is an identity matrix?

Answer 4

A matrix with 1s in the leading diagonals with all other elements as 0

Question 5

What must be true for matrices to be added or subtracted?

Answer 5

The matrices must have the same dimensions.

Question 6

What is a scalar?

Answer 6

A dimensional quantity - a number

Question 7

What law is A * (B * C) = (A * B) * C ?

Answer 7

The associative law

Question 8

What law is A * B = B * A

Answer 8

The commutative law

Question 9

Is matrix addition associative/commutative?

Answer 9

Yes, both

Question 10

Is matrix subtraction associative/commutative?

Answer 10

No. neither

Question 11

Is matrix multiplication commutative?

Answer 11

Not in general

Question 12

The composite matrix ABCD should be multiplied from…

Answer 12

… right to left

A(B(C(D)))

Question 13

What point always maps onto itself after a linear transformation?

Answer 13

The origin

Question 14

(k 0)

0 k

Answer 14

An enlargements by scale factor k

Question 15

(k 0)

0 1

Answer 15

Stretch scale factor k parallel to X axis

Question 16

(1 0)

0 k

Answer 16

Stretch scale factor k parallel to Y axis

Question 17

Rotation, anticlockwise by θ

Answer 17

(cos θ - sin θ)

sin θ cos θ

Question 18

G followed by H is …

Answer 18

H x G

Question 19

P followed by Q is …

Answer 19

Q x P

Question 20

(1 k)

0 1

Answer 20

A shear, x - axis fixed, with (0,1) mapped to (k,1)

Question 21

(1 0)

k 1

Answer 21

A shear, y - axis fixed, with (1,0) mapped to (1,k)

Question 22

A shear maintains…

Answer 22

area

Question 23

Invariant points are…

Answer 23

points which map to themselves

Question 24

How do you find points that are invariant under a transformation:
(a b)
(c d)

Answer 24

(a b) x (x) = (x)

c d) x (y) = (y

Question 25

What is a plane?

Answer 25

A plane is a flat two-dimensional surface that extends infinitely far.

Question 26

I3 with one -1 is a...

Answer 26

...reflection in a plane

Question 27

I3 with two -1 is a...

Answer 27

...rotation 180 around an axis

Question 28

What is an algorithm?

Answer 28

An algorithm is a FINITE sequence of operations for carrying out a procedure.

Question 29

What are the three conditions for an algorithm?

Answer 29

1. Must be unambiguous 2. Must be deterministic (no chance or randomness) 3. Must be finite

Question 30

How do you find the lower bound for bin packing?

Answer 30

Add all of the lengths together and divide by the bin-size

Question 31

First fit: remember to...

Answer 31

1. Write "saturated"

Question 32

First fit decreasing: remember to:

Answer 32

1. Write out the ordered elements in decreasing order | 2. Write "saturated"

Question 33

True or False: Full Bin strategy is the most efficient bin packing algorithm.

Answer 33

False, because full bin strategy is not an algorithm

Question 34

What is meant by a heuristic algorithm?

Answer 34

An algorithm that will usually find a good solution, although not necessarily an optimal solution.

Question 35

Quick sort: remember to

Answer 35

1. circle pivot | 2. underline sorted values

Question 36

What complexity does quick sort have?

Answer 36

quadratic complexity

Question 37

What is the maximum number of comparisons for quick sort?

Answer 37

0.5 * n^2 * n

Question 38

What is a minimum spanning tree?

Answer 38

A minimum spanning tree is the least weight connected graph that includes every vertex and contains no cycles

Question 39

What is the complex conjugate of a + bi

Answer 39

a - bi

Question 40

z = a + bi ... zz* = ...

Answer 40

zz* = a^2 + b^2

Question 41

What is the complex conjugate root theorem?

Answer 41

If a polynomial with real coefficients has a root z = a + bi, then z = a - bi is also a root. (Roots always come in pairs)

Question 42

What are the first 2 lines of working for the question... A quadratic equation has a root 3-2i. Find the quadratic equation

Answer 42

, if 3-2i us a root, then 3+2i is also a root bit the complex conjugate root theorem. (z - ( 3-2i )) ( z - ( 3+2i )) = 0

Question 43

If a quadratic equation has roots P and Q, write the quadratic equation.

Answer 43

z^2 - (P+Q)z + PQ = 0

Question 44

How do you square root 3+4i?

Answer 44

3+4i = (a+bi)^2 3 = a^2 - b^2 4 = 2ab ^equate real and imaginary parts Solve sim eqs.

Question 45

Multiplying by i... (in terms of argand diagram)

Answer 45

... rotates the number by 90 anticlockwise about the origin

Question 46

180 degrees in radians

Answer 46

pi

Question 47

2pi radians in degrees

Answer 47

360

Question 48

90 degrees in radians

Answer 48

pi/2

Question 49

What is the cartesian form of a complex number?

Answer 49

z = x + iy

Question 50

If z = x + iy, what is θ?

Answer 50

θ = arg(z) = arctan (y/x)

Question 51

If z = x + iy, what is r?

Answer 51

r = |z| = sqrt(x^2 + y^2)

Question 52

What is the range of arguments θ in modulus argument form?

Answer 52

-pi < θ <= pi aka the principal argument

Question 53

Write z in modulus argument form

Answer 53

z = r (cosθ + i sin θ)

Question 54

Multiplying two complex numbers together ...

Answer 54

... multiplies their moduli (length) and adds their arguments

Question 55

Multiplying two complex numbers together ...

Answer 55

... divides their moduli (length) and subtracts their arguments

Question 56

|z| = r represents ...

Answer 56

|z| = r represents a circle centred at the origin with radius r

Question 57

|z-a| = r represents ...

Answer 57

|z-a| = r represents a circle centred at 'a' with radius r

Question 58

arg(z-a) = θ represents the...

Answer 58

arg(z-a) = θ represents the locus of a half line of points from a measured θ from the positive horizontal axis. z=a is NOT a part of the locus

Question 59

|z-a| = |z-b| represents...

Answer 59

|z-a| = |z-b| represents all points which lie on the perpendicular bisector of a and b

Question 60

What two ways are there to solve |z-a| = |z-b|

Answer 60

1. Using the normal perpendicular bisector method | 2. equate equations of circle