LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE Examination – Mathematics

Third Semester – OCT/NOV 2010

MT 3501/MT 3500 – Algebra, Calculus and Vector Analysis

 

 

Date & Time:                                Dept. No.                                                    Max. : 100 Marks

 

 

PART – A

Answer ALL questions.                                                                                                  (10 ´ 2 = 20)

1. Evaluate
2. Find the Jacobian of the transformation x = u (1 + v) ; y = v (1 + u).
3. Find the complete solution of z = xp + yq + p² – q².
4. Solve
5. For , find div at (1, -1, 1)
6. State Green’s theorem.
7. What is L(f¢¢ (t))?
8. Compute
9. Find the sum and number of all the divisors of 360.
10. Define Euler’s function f(n) for a positive integer n.

 

PART – B

Answer any FIVE questions                                                                                          (5 ´ 8 = 40)

11. Evaluate by changing the order of the integration.
12. Express in terms Gamma functions.
13. Solve z²( p²+q² + 1 ) = b²
14. Solve p² + q² = z²(x + y).
15. Find
16. Find
17. Prove that
18. Show that 18! + 1 is divisible by 437

 

PART – C

Answer any THREE questions.                                                                               (2 ´ 20 =40)

19. (a) Evaluate  taken through the positive octant of the sphere x² + y² + z² = a².

(b)  Show that

20. (a) Solve (p² + q²) y = qz.

(b)  Solve (x² – y²)p + (y² – zx)q = z² – xy

21. (a) Verify Gauss divergence theorem for taken over the region bounded by the planes x = 0, x = a, y = 0 y = a, z = 0 and z = a.

(b)  State and prove Fermat’s theorem.

22. (a) Using Laplace transform solve  given that .

(b)  Show that if n is a prime and r < n, then (n – r)!  (r – 1)! + (-1)^{r – 1} º 0 mod n.

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