SEM 2

Question 1

Can the hcf be negative

Answer 1

Yes

Question 2

What is the prime number formula (Works up to n = 41)?

Answer 2

P(n) = n^2 - n + 41

Question 3

What are the steps of mathematical induction?

Answer 3

Prove P(N) is true

Prove P(k+1) is true given P(k) is true

Question 4

Strong induction?

Answer 4

Like mathematical except

P(k + 1) is true when P(N), P(N + 1) … is true

Question 5

What is fermats little theorem?

Answer 5

a^p = a mod p

a^p-1 = 1 mod p (if p is not factor of a)

Question 6

Key generation: n = ?

Answer 6

pq

Question 7

Key Gen: e = ?

Answer 7

e is between 1 and (p-1)(q-1) and

e and (p-1)(q-1) are coprime

Question 8

Key gen: d = ?

Answer 8

d > 0 such that

de == 1 mod (p-1)(q-1)

Question 9

What is the public key?

Answer 9

(n,e)

Question 10

What is private key?

Answer 10

d

Question 11

How to encrypt?

Answer 11

For each value M

c = M^e mod n

Question 12

How to decrypt?

Answer 12

M = c^d mod n

Question 13

How to find inverse of 2x2 matrix?

Answer 13

1/ad - bc (d, -b, -c, a)

Question 14

How to find inverse of any matrix (not 2x2)

Answer 14

• Write identity matrix next to it
• Turn left side into identity matrix and right side is inverse

Question 15

What is minor?

Answer 15

Determinant (without working out result)

Question 16

What is cofactor?

Answer 16

Multiply minor by -1

Question 17

When does a system of linear equations have a unique solution?

Answer 17

When determinant != 0

Question 18

When is a matrix invertible?

Answer 18

When the determinant != 0

Question 19

If a matrix is diagonal what are its eigenvalues?

Answer 19

The diagonal elements

Question 20

What makes a matrix similar?

Answer 20

B = P^-1 A P

Question 21

Do similar matrices have the same characteristic polynomials and eigenvalues?

Answer 21

Yes

Question 22

What makes A diagonalisable?

Answer 22

If there is a diagonal matrix which is similar

Question 23

Cartesian product?

Answer 23

Set of ordered pairs

Question 24

Permutation? P(n,k)

Answer 24

n! / (n-k)!

• ordered collection

Question 25

Combinations? C(n,k)

Answer 25

n! / k! (n-k)!

Question 26

Probability space consists of?

Answer 26

Sample space and probability function

Question 27

Conditional probability? A given B

Answer 27

P(A n B) / P(B)

Question 28

Bayes theorem? (A given B)

Answer 28

P(B | A)P(A) / P(B)

Question 29

Law of total probability?

Answer 29

P(A) = P(A|B1)P(B1) + … + P(A|Bn)P(Bn)

Question 30

Expectation value?

Answer 30

Sum of:

x * p(x)

Question 31

Var?

Answer 31

E(X^2) - E(X)^2

Question 32

sd?

Answer 32

Root varience

Question 33

Bernoulli distribution?

Answer 33

• Range of possible values 0 -1
• P(x = 1) = p
• P(X = 0) = 1- p
• E(X) = p
• var(X) = p(1-p)

Question 34

Binomial distribution

Answer 34

• Range up to n
• P(X = k) = C(n,k)p^k(1-p)^n-k for each k
• E(X) = np
• var(X) = np (1-p)

Question 35

Poisson distribution?

Answer 35

• Unlimited range
• P(X = k) = (lambda)^k/ k! * e^ -(lambda)
• E(X) and var(X) = lambda

Question 36

Difference eqution?

Answer 36

vk+1 = Avk

Question 37

Solve difference equation?

Answer 37

vk = A^k v0

Question 38

Markov process?

Answer 38

• Calculate prop for event a and event b using given info
• Place in Mx = b (each value matrix)
• For given time distribution is given by M^k x

Question 39

Transition matrix?

Answer 39

Probability of moving from state to state as a vector

Question 40

Markov process (Long term behavior)?

Answer 40

vk = v

• v is (lambda)-eigenvector of M