2013

(November)

MATHEMATICS

(Major)

Course: 501

(Logic and Combinatorics, and Analysis – III)

Full Marks: 80

Pass Marks: 32

Time: 3 hours

The figures in the margin indicate full marks for the questions

(A) Logic and Combinatorics

(Marks: 35)

1. (a) Define truth function. 1

(b) Letbe ‘it is cold’ andbe ‘it is raining’. Give verbal sentence which describes each of the following: 2

(c) Construct the truth table for. State whether it is a tautology or not. 3

(d) Prove that every truth function can be generated by. Can you generate a truth function by using and only? 4

Or

Give the arithmetic representation of the form. Also show that

2. (a) What do you mean by equivalent statements? 1

(b) Write the rule p and rule t. 2

(c) Translate into symbols 3

- Not all birds can fly.
- Anyone can do it.
- Some people are intelligent.

(d) Derive any one of the following: 4

- Everyone who buys a ticket receives a prize. Therefore, if there is no prize, there nobody buys ticket.
- All men are mortal. Ram is a man. Hence Ram is mortal.

3. (a) State the Pascal’s identity. 1

(b) Find the coefficient of in. 2

(c) Define Ramsey number. Prove that. 4

Or

Define Catalan numbers. Prove that Catalan number

4. (a) State the pigeonhole principle. 1

(b) How many integers between 100 and 700 are divisible by 3 or 5? 3

(c) Prove that given any 12 natural numbers we can choose 2 of them such that their difference is divisible by 11. 4

Or

Define binomial generating function. Find both binomial and exponential generating functions for the sequence 2, 2, 2, 2, ....

(B) Analysis – III (Complex Analysis)

(Marks: 45)

5. (a) State the condition under which a function is said to be analytic. 1

(b) Define harmonic function. Show that is harmonic. 3

(c) State and prove the necessary conditions for a functionto be analytic at all points in a region. 6

Or

Show that

is not analytic at the origin, although Cauchy-Riemann equations are satisfied. What is your opinion in this case?

6. (a) Define Jordan’s arc.

(b) Find the value of the integral

where .

(c) State and prove Cauchy’s integral theorem.

(d) If a function is analytic for all finite values of and is bounded, then show that it is constant. 6

Or

Evaluate:

- , whereis
- , whereis

7. (a) State and prove Taylor’s series. 1+5=6

(b) Expand is Laurent’s series, where. 2

8. (a) Define an isolated singular point of a function . 1

(b) Discuss the singularity of

at 2

(c) Evaluate (any two): 5x2=10

- , where is

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