LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc., DEGREE EXAMINATION – MATHEMATICS

THIRD SEMESTER – NOVEMBER 2012

MT 3501/3500 – ALGEBRA, CALCULUS AND VECTOR ANALYSIS

 

 

 

Date : 02-11-2012              Dept. No.                                        Max. : 100 Marks

Time : 9.00 – 12.00

 

PART – A

 

ANSWER ALL THE QUESTIONS:                                                                                                (10 x 2 =  20)

 

1. Evaluate .
2. Evaluate .
3. Eliminate the arbitrary constants from .
4. Find the complete solution for
5. Find , if .
6. Prove that div , where is the position vector.
7. Find L(Sin2t).
8. Find .
9. Find the number and sum of all the divisors of 360.
10. State Fermat’s theorem.

 

PART – B

ANSWER ANY FIVE QUESTIONS:                                                                                  (5 x 8 = 40)

11. Change the order of integration and evaluate the integral  .
12. Express  in terms of Gamma functions and evaluate the integral .
13. Solve
14. Solve
15. Find .
16. Find .
17. Show that
18. Show that is divisible by 22.

 

PART – C

 

ANSWER ANY TWO QUESTIONS                                                                                               (2x 20 = 40)

 

19. (a) Evaluate taken over the positive quadrant of the circle .

(b)  Prove that

20. (a) Solve

 

(b) Solve (y+z)p + (z+x)q = x+y.

 

21.(a)  Verify Stoke’s theorem for  taken over the upper half surface of

the  sphere  x²+y² +z² = 1, z 0 and the boundary curve C, the x²+y²  = 1, z=0.

 

(b)  State and prove Wilson’s Theorem.

22. Using Laplace transform solve the equation given y(0) = 0 , y¹(0)= -1.

 

 

 

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