Brainscape's Knowledge GenomeTM

Browse over 1 million classes created by top students, professors, publishers, and experts.


See full index

Further Maths A Level > CAP 1 > Flashcards

CAP 1 Flashcards

1
Q

What is a matrix?

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
2
Q

How many rows and columns does a MxN matrix have?

A

M rows and N columns

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
3
Q

What is a square matrix

A

A matrix that has the same number of rows as columns

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
4
Q

What is an identity matrix?

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
5
Q

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

A

The matrices must have the same dimensions.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
6
Q

What is a scalar?

A

A dimensional quantity - a number

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
7
Q

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

A

The associative law

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
8
Q

What law is A * B = B * A

A

The commutative law

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
9
Q

Is matrix addition associative/commutative?

A

Yes, both

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
10
Q

Is matrix subtraction associative/commutative?

A

No. neither

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
11
Q

Is matrix multiplication commutative?

A

Not in general

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
12
Q

The composite matrix ABCD should be multiplied from…

A

… right to left

A(B(C(D)))

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
13
Q

What point always maps onto itself after a linear transformation?

A

The origin

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
14
Q

(k 0)

0 k

A

An enlargements by scale factor k

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
15
Q

(k 0)

0 1

A

Stretch scale factor k parallel to X axis

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
16
Q

(1 0)

0 k

A

Stretch scale factor k parallel to Y axis

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
17
Q

Rotation, anticlockwise by θ

A

(cos θ - sin θ)

sin θ cos θ

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
18
Q

G followed by H is …

A

H x G

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
19
Q

P followed by Q is …

A

Q x P

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
20
Q

(1 k)

0 1

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
21
Q

(1 0)

k 1

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
22
Q

A shear maintains…

A

area

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
23
Q

Invariant points are…

A

points which map to themselves

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
24
Q

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

A

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

c d) x (y) = (y

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
25
Q

What is a plane?

A

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

26
Q

I3 with one -1 is a…

A

…reflection in a plane

27
Q

I3 with two -1 is a…

A

…rotation 180 around an axis

28
Q

What is an algorithm?

A

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

29
Q

What are the three conditions for an algorithm?

A
  1. Must be unambiguous
  2. Must be deterministic (no chance or randomness)
  3. Must be finite
30
Q

How do you find the lower bound for bin packing?

A

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

31
Q

First fit: remember to…

A
  1. Write “saturated”
32
Q

First fit decreasing: remember to:

A
  1. Write out the ordered elements in decreasing order

2. Write “saturated”

33
Q

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

A

False, because full bin strategy is not an algorithm

34
Q

What is meant by a heuristic algorithm?

A

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

35
Q

Quick sort: remember to

A
  1. circle pivot

2. underline sorted values

36
Q

What complexity does quick sort have?

A

quadratic complexity

37
Q

What is the maximum number of comparisons for quick sort?

A

0.5 * n^2 * n

38
Q

What is a minimum spanning tree?

A

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

39
Q

What is the complex conjugate of a + bi

A

a - bi

40
Q

z = a + bi …

zz* = …

A

zz* = a^2 + b^2

41
Q

What is the complex conjugate root theorem?

A

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)

42
Q

What are the first 2 lines of working for the question…

A quadratic equation has a root 3-2i. Find the quadratic equation

A

, 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

43
Q

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

A

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

44
Q

How do you square root 3+4i?

A

3+4i = (a+bi)^2

3 = a^2 - b^2
4 = 2ab
^equate real and imaginary parts

Solve sim eqs.

45
Q

Multiplying by i… (in terms of argand diagram)

A

… rotates the number by 90 anticlockwise about the origin

46
Q

180 degrees in radians

A

pi

47
Q

2pi radians in degrees

A

360

48
Q

90 degrees in radians

A

pi/2

49
Q

What is the cartesian form of a complex number?

A

z = x + iy

50
Q

If z = x + iy, what is θ?

A

θ = arg(z) = arctan (y/x)

51
Q

If z = x + iy, what is r?

A

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

52
Q

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

A

-pi < θ <= pi

aka the principal argument

53
Q

Write z in modulus argument form

A

z = r (cosθ + i sin θ)

54
Q

Multiplying two complex numbers together …

A

… multiplies their moduli (length) and adds their arguments

55
Q

Multiplying two complex numbers together …

A

… divides their moduli (length) and subtracts their arguments

56
Q

|z| = r represents …

A

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

57
Q

|z-a| = r represents …

A

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

58
Q

arg(z-a) = θ represents the…

A

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

59
Q

|z-a| = |z-b| represents…

A

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

60
Q

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

A
  1. Using the normal perpendicular bisector method

2. equate equations of circle

Further Maths A Level (1 decks)

Brainscape helps you reach your goals faster, through stronger study habits.
© 2024 Bold Learning Solutions. Terms and Conditions