## Wednesday, December 26, 2018

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) Let be ‘it is cold’ and be ‘it is raining’. Give verbal sentence which describes each of the following: 2
1. 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
1. Not all birds can fly.
2. Anyone can do it.
3. Some people are intelligent.
(d) Derive any one of the following: 4
1. Everyone who buys a ticket receives a prize. Therefore, if there is no prize, there nobody buys ticket.
2. 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 function to 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:
1. , where is 2. , where is 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
1. 2. 3. , where is 4. ***