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
(10  2 = 20)
Answer ALL questions.

2 a
  drd
1. Evaluate
0 0
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
2 z
y 2
 sin y
5. For f  xy 2i  2 x 2 yzj  3 yz 2k , find div f at (1, -1, 1)
6. State Green’s theorem.
7. What is L(f (t))?


1
8. Compute L1 
 ( s  2)2  16 


9. Find the sum and number of all the divisors of 360.
10. Define Euler’s function (n) for a positive integer n.
PART – B
(5  8 = 40)
Answer any FIVE questions
4a 2 ax
11. Evaluate by changing the order of the integration
 
0 x
1
12. Express
x
m
(1  x n ) p dx in terms Gamma functions.
0
2
13. Solve z ( p2+q2 + 1 ) = b2
14. Solve p2 + q2 = z2(x + y).
 cos 3t  cos 2t 
15. Find L 

t


2
4a
dydx .


s2
16. Find L1 
 ( s  4)( s 2  1) 


 
17. Prove that  r n  n(n  1)r n  2
18. Show that 18 + 1 is divisible by 437
PART – C
(2  20 =40)
Answer any THREE questions.
19. (a) Evaluate
 xyz dxdydz taken through the positive octant of the sphere x2 + y2 + z2 = a2.
(b) Show that   m, n  
(m)(n)
 ( m  n)
20. (a) Solve (p2 + q2) y = qz.
(b) Solve (x2 – y2)p + (y2 – zx)q = z2 - xy
21. (a) Verify Gauss divergence theorem for F  xiˆ  y ˆj  zkˆ 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
d2y
dt 2
5
dy
dy
 6 y  e t given that y  0;  1 when t  0 .
dt
dx
(b) Show that if n is a prime and r < n, then (n – r)! (r – 1)! + (-1)r – 1  0 mod n.