2015
(May)
MATHEMATICS
(Major)
Course: 402
(Linear Programming and Analysis  II)
Full Marks: 80
Pass Marks: 32
Time: 3 hours
The figures in the margin indicate full marks for the questions
GROUP – A
(Linear Programming)
(Marks: 45)
1. (a) How many components are essential to formulate any LP Problem? 1
(b) Write down the general linear programming problem with decision variables and constraints. 2
(c) Evening shift resident doctors in a Government hospital work five consecutive days and have two consecutive daysoff. Their five days of work can start on any day of the week and the schedule rotates indefinitely. The hospital requires the following minimum number of doctors working:
Sun

Mon

Tues

Wed

Thurs

Fri

Sat

35

55

60

50

60

50

45

No more than 40 doctors can start their five working days on the same day. Formulate a general linear programming model to minimize the number of doctors employed by the hospital. 3
(d) The graphical method to solve the following LP problem: 4
Maximize
Subject to the constraints
And
2. (a) What type of variables needed to add in the given LP Problem to convert it into standard form when the constraints is ‘’ type? 1
(b) Write down the key column from the following table: 2
0

8

2

3

0

1

0

0
 
0

10

0

2

5

0

1

0
 
0

15

3

2

4

0

0

1
 
3

5

4

0

0

0

(c) Solve by simplex method: 5
Maximize
Subject to the constraints
And
(d) Solve by twophase method: 7
Minimize
Subject to constraints
And
Or
Solve by BigM method:
Maximize
Subject to the constraints
And
3. (a) State fundamental duality theorem. 1
(b) What should be the type of solution of a dual problem who’s primal has an unbounded solution? 1
(c) Write down the two rules for constructing the dual problem from primal. 2
(d) Find the dual of the following LP problem: 4
Maximize
Subject to the constraints
And
Or
“If the kth constraint of a primal be an equation, then the kth dual variable will be unrestricted in sing.” Prove it.
4. (a) Fill in the blank:
In a balanced transportation problem having origins and destinations, the exact number of basic variables is ____. 1
(b) Write down the two properties of a loop in a transportation problem. 2
(c) Obtain the initial basic feasible solution to the following transportation problem by Vogel’s approximation method and prove that the solution is degenerate: 9
4

6

5

2

6
 
6

4

1

4

10
 
5

2

3

1

12
 
4

6

7

8

14
 
9

16

10

7

42

Or
Discuss the ‘MODI’ method to test the optimality of a solution to a transportation problem. 9
GROUP – B
[Analysis – Ii (Multiple Integral)]
(Marks: 35)
5. (a) Fill in the blank: 1
The integral of a periodic function over any interval whose length is equal to its period always has the ____.
(b) The function is periodic with period on the interval. Find its Fourier series. 4
(c) Find the Fourier series of the periodic function with period, defined as 5
Also find the sum of series at and
Or
Expand the function
in the interval as a sine series.
6. (a) Define a plane curve. 1
(b) Evaluate the integral where is the curve 2
(c) Prove that every continuous function in is integrable. 4
(d) State and prove Green’s theorem. 1+5=6
Or
Change the order of integration
and also prove that where 6
7. (a) State Gauss’ theorem. 1
(b) Express a surface integral in terms of a double integral. 2
(c) Find the volume of the sphere using polar coordinates. 3
(d) Prove Stokes theorem. 6
Or
Compute the volume of the ellipsoid