2014
(May)
MATHEMATICS
(Major)
Course: 402
[A: Linear Programming, B: Analysis – II
Multiple Integral]
Full Marks: 80
Pass Marks: 32
Time: 3 hours
The figures in the margin indicate full marks for the questions
A: LINEAR PROGRAMMING
(Marks: 45)
1. (a) Is the solution 1
a basic solution of the equations
(b) Show that a hyperplane is a convex set.
(c) A factory is engaged in manufacturing two products – A and B which involve lathe work, grinding and assembling. The cutting, grinding and assembling times required for one unit of A and 2, 1 and 1 hours respectively and for one unit of B are 3, 1 and 3 hours respectively. The profits on each unit of A and B are Rs. 2 and Rs. 3 respectively. Assuming that 300 hours of lathe time, 300 hours of grinding time and 240 hours of assembling time are available, formulate a linear programming problem in terms of maximizing the profit on the items manufactured.
(d) Use the graphical method to solve the following LP problem:
Minimize
Subject to the constraints
2. (a) Write ‘true’ or ‘false’: “If a linear programming problem has a feasible solution, then it has a basic feasible solution.”
(b) Express the following LPP in the standard form by adding additional variables to the left side of each constraint and assign a zerocost coefficient to these in the objective function:
Maximize 2
Subject to the constraints
(c) Solve by simplex method: 5
Maximize
Subject to the constraints
(d) Solve by twophase method: 7
Minimize
Subject to the constraints
Or
Solve by BigM method:
Minimize
Subject to the constraints
3. (a) Fill in the blank: If any of constraints in the primal problem is a perfect equality, then the corresponding dual variable is ____. 1
(b) Write ‘true’ or ‘false’: “An LPP has a finite optimal solution if and only if there exist feasible solutions to both the primal and dual problems.”
(c) Find the dual of the following primal problem:
Minimize
Subject to the constraints
(d) Write down the correspondence rules between the primal problem and the dual problem.
Or
If the dual problem has no feasible solution and the primal problem has a feasible solution, then prove that the primal objective function is unbounded.
4. (a) What do you mean by rim condition of a transportation problem? 1
(b) Examine with diagram whether the following ordered set of cells forms a loop or not: 2
(c) Obtain an optimal solution to the following transportation problem by the MODI method: 9
Supply
 
16

20

12

200
 
14

8

18

160
 
26

24

16

90
 
Demand

180

120

150

450

Or
Write short notes on: 4 ½ x2=9
 Least cost method.
 Vogel’s approximation method.
B: ANALYSIS – II
(Multiple Integral)
(Marks: 35)
5. (a) Write ‘true’ or ‘false’: “A periodic function of bounded variation can be expressed as a Fourier series.” 1
(b) Find the Fourier coefficient (for odd) of the following function with period 2
(c) Find the Fourier series generated by the periodic function of period. 3
(d) Find the trigonometrically series which converges in to the function 4
Or
Expand the function in a Fourier series in
6. (a) Define line integral in twodimensional space. 1
(b) Evaluate the integral taken along the line segment from to . 2
(c) Prove that a bounded function on a region, having an infinite number of discontinuities lying on a finite number of smooth curves, is integrable on 5
Or
Evaluate over the part of the plane bounded by the lines and the parabola.
(d) With the help of Green’s theorem, prove that the line integral
taken in the positive direction over any closed contour with the origin inside it, is equal to . 5
Or
Compute the double integrals of the function over
7. (a) State Stokes’ theorem. 1
(b) Find the length of the curve,, , 2
(c) Compute the surface area of the sphere 4
Or
Show that
is equal to 3/8, where S is the outer surface of the sphere in the 1st octant.
(d) State and prove Gauss’ theorem. 5
Or
Evaluate the surface integral by Gauss’ theorem of the following:
Over the sphere