## Monday, December 31, 2018

2013
(May)
MATHEMATICS
(Major)
Course: 402
(A: Linear Programming, B: Analysis – II)
Full Marks: 80
Pass Marks: 32
Time: 3 hours
The figures in the margin indicate full marks for the questions

1. (a) How many basic assumptions are necessary for all linear programming models? 1
(b) Write the general linear programming problem with decision variables and constraints. 2
(c) A company makes 3 models of calculator – A, B and C at factory I and factory II. The company has orders for at least 6400 calculators of model A, 4000 calculators of model B and 4800 calculators of model C. At factory I, 50 calculators of model A, 50 of model B and 30 of model C are made everyday. It costs Rs. 12,000 and Rs. 15,000 each day to operate factory I and II respectively. Find the number of days each factory should operate to minimize the operating costs and still meet the demand. 3
(d) Solve graphically the following LPP: 4
Maximize Subject to the constraints 2. (a) For what reasons, additional variables are to be addend to convert an LPP into standard form? 2
(b) Solve by simplex method: 5
Maximize Subject to the constraints (c) Solve by two-phase method: 8
Maximize Subject to the constraints Or
Solve by Big-M method:
Maximize Subject to the constraints 3. (a) What happens in the dual if the variables in primal in unrestricted in sign? 1
(b) Fill up the blank: 1
“If the primal problem has an unbounded objective function, then the dual has ____ solution.”
(c) Write two rules for constructing the dual from the primal. 2
(d) Find the dual of the following primal LPP: 4
Minimize Subject to the constraints  Unrestricted in sign.
Or
Prove that the dual of the dual linear programming problem is the primal.
4. (a) What is the necessary and sufficient condition for the existence of a feasible solution to the transportation problem? 1
(b) Mention two properties of a loop in a transportation problem. 2
(c) Obtain an optimal solution to the following transportation problem by Vogel’s method: 9    Supply 19 30 50 10 7 70 30 40 60 9 40 8 70 20 18 Demand 5 8 7 14 34

Or
What do you mean by balanced transportation problem? Find the initial basic feasible solution of the following problem with the help of least cost method:    Supply 1 2 1 4 30 3 3 2 1 50 4 2 5 9 20 Demand 20 40 30 10

B: ANALYSIS – II
(Multiple Integral)
(Marks: 35)

5. (a) When will the trigonometric series be a Fourier series? 1
(b) For a periodic function of period , show that 2 Where is any number?
(c) If the function is periodic with period on the interval , then find the Fourier series of .    3
(d) Obtain the Fourier series of the periodic function with period defined as 4 And hence deduce that Or
If a function is bounded, periodic with period and integrable on and piecewise monotonic on , then prove that  Where , are Fourier coefficients of .
6. (a) Write the parametric representation of the curve 1
(b) If a function is defined as Then show that 3

(c) Evaluate Over the region 4
Or
Evaluate Over the region Where (d) State and prove Green’s theorem. 5
Or
Using Green’s theorem, compute the difference between the line integrals. And Where and are respectively the straight line and the parabolic are , joining the points and .
7. (a) Define a surface in 1
(b) State Gauss’ theorem. 1
(c) Evaluate 2
Where is the outer side of the part of the sphere 4
Or
Find the volume of the solid bounded by the surface and the plane (d) Using Stokes’ theorem, show that 5 Where is the portion of the surface Or
Using Gauss’ theorem, show that Where is the closed surface bounded by the cone and the plane and , , are direction cosines of the outward drawn normal of .

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