MATHEMATICS-HONOURS

BangabasiEveningCollege

Part-II

Third Paper (Module V and VI)

Under 1+1+1 Systems

Group-A

Modern Algebra-II

Answer any two Questions

1. (a) Define a cyclic group. Give an example of an infinite cyclic group.

(b) Define order of a permutation. Determine the order of the permutation.

2. Prove that a finite integral domain is a field.

3. Examine if the ring of matrices is field.

4. Prove that the order of a subgroup of finite group divides the order of the group.

Group-B

Linear Programming Problem and Game Theory

Answer any two Questions

4. (a) Two players, A and B, play the following game : B hides a 5 rupee coin under one of the 2 tins he has. lf A correctly guesses the tin that contains the coin, she gets the coin. If her guess is wrong, she pays Rs. 2 to B. Formulate this game as a matrix game. Solve the game and find the optimal strategies for both the players, as well as the value of the game.

(b) While solving an LPP by the Simplex method, the optimal table obtained is given below:

CB / Cj
Basic Variables / 1
x1 / 2
x2 / 0
x3 / 0
x4 / 0
x5 / Solution
2 / x2 / 0 / 1 / ¼ / ¼ / 0 / 5
1 / x1 / 1 / 0 / -½ / ½ / 0 / 2
0 / x5 / 0 / 0 / ¾ / -¼ / 1 / 6
/ 0 / 0 / 0 / 1 / 0 / 12

(i) Find an alternative optimal basic feasible solution.

(ii) Find an alternative optimal non-basic feasible solution. 3+4

5. (a) Find all the basic solutions of the following system:
4x1 + 2x2 + x3=4
2x1 + x2 + 5x3=5

(b) Sketch the feasible region given by the following constraints:

2x1 + x2 ≥ 1
x1 - x2 ≥ 0
-x1 + 2x2 ≤ 2
x1,x2 > 0

Hence find the minimum and maximum values of the objective function Z=2x1 + x2 over this feasible region. 3+4

6. Find an initial basic feasibility solution of the following transportation problem, using the VAM method, and hence solve the problem.

D1 / D2 / D3 / D4 /
O1 / 19 / 30 / 50 / 10 / 7
O2 / 70 / 30 / 40 / 60 / 9
O3 / 40 / 8 / 70 /
  1. 20
/ 18
/ 5 / 8 / 7 / 14

7. (a) Reduce the following Pay-off matrix to 2x2 matrix by dominance property and thus solve the Game problem.

B1 / B2 / B3 / B4
A1 / 4 / 2 / 3 / 2
A2 / -2 / 4 / 6 / 4
A3 / 2 / 1 / 3 / 5

(b) Use the theory of linear programming to solve the game problem whose Pay-off matrix is

B1 / B2 / B3
A1 / 1 / 0 / -2
A2 / 0 / 3 / 2

4+3

8. Solve the following assignment problem with the following cost matrix:

M1 / M2 / M3 / M4 / M5
I / 5 / 11 / 10 / 12 / 4
II / 2 / 4 / 6 / 3 / 5
III / 3 / 12 / 5 / 14 / 6
IV / 6 / 14 / 4 / 11 / 7
V / 7 / 9 / 8 / 12 / 5

And find the minimum cost of assignment.

Group-C

Analysis-II

Full Marks-11

Answer any one Question

9. (a) State and Prove Leibnitz’s Test on infinite series of real numbers.

(b) Test for convergence of the following series: . Is it is absolute convergent.

(c) Use MVT; find the value of 5+3+3

10. (a) Prove that a=120, b=60, c=180 if

(b) Let be a function such that

(i) f is continuous on [a, b]

(ii) f is derivable in (a, b)

and (iii) f(a)=f(b).

Prove that there exists such that.

(c) If, then prove that,

, according as n is even or odd. 4+4+3

Group-D

Differential Equations I

Full Marks-15

Answer any three Questions (3x5=15)

11. Consider the change of independent variable,,for the equation

.

Show that the equation in z is

.

12. Show that is a solution of the equation

for , then find the general solution.

13. (a) State a theorem on existence of solution of first order and first degree differential equation.

(b) Prove that the system of confocal conics is self-orthogonal where is a parameter. (2+3)

14. Solve the given equation by converting it to a system.

,where.

15. Find the eigen values and eigen functions of

;.

16. Solve by Lagrange’s method, the equation

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