## Wednesday, December 26, 2018

2015
(May)
MATHEMATICS
(Major)
Course: 604
Full Marks: 80
Pass Marks: 32
Time: 3 hours
The figures in the margin indicate full marks for the questions.
[ (a) Financial Mathematics
(b) Operations Research. ]
(a) Financial Mathematics
(Marks: 45)

1. (a) Describe briefly what you mean by market equilibrium. 2
(b) The supply set consists of the points such that and the demand set consists of the points such that . An excise tax of Rs. 2 per unit is imposed. Determine the new equilibrium price and quantity. 3
2. Describe the economic interpretation of cobweb model. 5
Or
The supply and demand sets are given by and . Find the price sequence.
3. Write a note on profit maximization. 5
Or
State briefly about optimization in an interval.
4. (a) Define elasticity of demand. 3
(b) For an efficient small farm, define start-up point and show that at start-up point, marginal cost is equal to average variable cost. 3
(c) For an efficient small farm, define break-even point. Show that at break-even point the derivative of average cost is zero. 4
5. (a) State True or False: Every critical point is a maximum. 1
(b) Find and classify the critical points of 4
(c) Find the maximum values of 5
6. (a) Define a technology matrix. 2
(b) Explain a two-industry economy with an example. 4
(c) The matrix for an investor is 4
Show that the portfolio is riskless.
What return is the investor guaranteed?

(b) Operations Research
(Marks: 35)

7. State briefly about operations research techniques. 5
Or
Write a note on scopes and limitations of operations research.
8. (a) Prove that in an assignment problem if we add or subtract a constant to every element of a row (or column) in the cost matrix, then an assignment which minimizes the total cost on one matrix, also minimizes the total cost on the other matrix. 3
(b) Find the optimal assignment and the corresponding assignment cost from the following cost matrix: 7
A B C D E
 1 2 3 4 5 9            8 7            6 4 5            7 5            6 8 8            7 6            3 5 8            5 4            9 3 6            7 6            8 5

Or
Find the optimal assignment profit from the following profit matrix:
I II III IV
 A B C D 42     35  28 21       30     25  20 15       30     25  20 15       24     20  16 12

9. (a) Who developed the dynamic programming technique? 1
(b) State what you mean by principle of optimality. 2
(c) Using dynamic programming, solve the following : 7
Maximize Subject to Or
Maximize Subject to   10. (a) State whether the following statement is True or False: 1
Integer programming is a special class of linear programming problem in which decision variables have integer solution.
(b) Solve the following by Gomory technique: 9
Maximize Subject to  Or
Maximize Subject to GROUP – B
[ (A) Space Dynamics
(b) Relativity]
(a) Space Dynamics
(Marks: 40)

If is the polar triangle of the triangle , then
1. 2. 3. 1. State True or False:
A general spherical triangle has more then one right angle.
1. Fill up the Gap:
In spherical triangle , the circular parts are ____.
2. If one triangle is the polar triangle of another triangle, then prove that the later is the polar of the former.    2
3. State and prove the cosine formula of a spherical triangle. 5
Or
In any spherical triangle , prove that 4. In an equilateral spherical triangle , prove that 3
Or
In a spherical triangle , and is the middle point of , then show that 5. If and are respectively the equatorial system and ecliptic system of coordinates of a star, then prove that and where is the obliquity of the ecliptic. 3+3=6
6. Define the following: 2x2=4
1. Annual motion of the sun.
2. Rising and setting of stars.

7. Write short notes on the following (any two): 2x2=4
1. Diurnal motion of heavenly bodies.
2. Hour angle and parallactic angle of a star.
3. Cardinal points and equinotical points.
8. If is the angle which a star makes at rising with the horizon, prove that 3 Where and have their own meanings.
9. State three Kepler’s laws. 3
10. Define the following (any two): 1x2=2
1. Aphelion.
2. Eccentric anomaly.
3. True anomaly.
11. Derive an expression for the position of a body in an elliptic orbit. 5
Or
If and are linear velocities of a planet at perihelion and aphelion respectively, prove that (b) Relativity
(Marks: 40)

12. What is an inertial frame of reference? Can earth be considered as an inertial frame of reference in true sense? Justify. 1+1=2
13. When does Lorentz transformation reduce to Galilean transformation? 1
14. Choose the correct answer: 1
If the interval of occurrence of two events is space-like, then there exists a frame of reference in which two events occur
1. At the same time.
2. At a finite interval of time.
3. At an infinite interval of time.
15. State the two postulates of special theory of relativity. 2
16. Write short notes on the following: 3+2=5
1. Length contraction.
2. Relative simultaneity.
17. (a) A particle with a mean proper lifetime of 2 sec moves through the laboratory with a speed of 0.9c. Calculate its lifetime as measured by an observer in the laboratory. 3
(b) The length of a rocket ship is 100 m on the ground. When it is in flight, its length observed on the ground is 99 m. calculate its speed. 3
(c) A scientist observes that a certain atom A moving relative to him with velocity emits a particle B which moves with velocity with respect to the atom. Calculate the velocity of the emitted particle relative to the scientist. 3
18. Establish the relation of variation of mass with velocity. Where is the velocity of the body when its mass is and is the mass of body at rest? 6
Or
Establish with usual notations.
19. What is the increase in the relativistic mass of a particle of rest mass 1 g when it is moving with 0.8c velocity? Hence, find its kinetic energy. 2
20. Answer any two questions: 6x2=12
1. Find the transformation laws of density in relativistic mechanics.
2. Deduce Lorentz transformation equations.
3. Calculate the velocity of an electron having a total energy of 2 MeV. ***