## Tuesday, January 01, 2019 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 days-off. 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 two-phase method: 7
Minimize Subject to constraints And Or
Solve by Big-M 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 k-th constraint of a primal be an equation, then the k-th 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 ***