LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – MATHEMATICS
THIRD SEMESTER – APRIL 2008
MT 3501 - ALGEBRA, CALCULUS AND VECTOR ANALYSIS
Date : 26-04-08
Time : 1:00 - 4:00
Dept. No.
XZ 8
Max. : 100 Marks
SECTION – A
Answer ALL questions.:
1. Evaluate
1
1
1
0
0
0
(10 x 2 = 20 marks)
   x dx dy dz .
2. If x  r cos  ; y  r sin , find the Jacobian of x and y with respect to r and  .
2 z
 sin x
xy
Find the complete solution of
z  px  9 y  p 2  92 .
Find  at (2,0,1) for  ( x, y, z )  x 2  3 y 2  z 2  6 .
State Stoke’s theorem.
Evaluate  (Sinh at).
 6 
Evaluate -1  ( S  2) 4  .


3. Solve
4.
5.
6.
7.
8.
9. Find the sum of all divisors of 360.
10. Compute  (720).
SECTION – B
Answer any FIVE questions.
11. By the changing the order of integration evaluate
  y
e
0 x y dy dx
(5 x 8 = 40 marks)
1
12. Express
m
n p
 x (1  x ) dx interms of Gamma function and evaluate
0
1
 x (1  x )
5
3 10
dx .
0
z
x y
   pq .
13. Obtain the complete and singular solutions of
pq q p
14. Solve z  p 2  q 2 .
15. Find  if   (6 xy  z 3 )i  (3x 2  z ) j  3( xz 2  y)k.
 cos 3t  cos 2t 
16. Evaluate (i)  ( sin 2 t ) (ii)  

t




2s 2  10s
17. Find   2

 ( s  2s  5)( s  1) 
18. Show that if x and y are both prime to the prime n, then xn-1-yn-1 is divisible by n. Deduce that x12y12 is divisible by 1365.
-1
1
SECTION – C
Answer any TWO questions.
(2 x 20 = 40 marks)
x y z
19. a) Evaluate    dx dy dz over the tetrahedron bounded by the planes    1 and the
a b c
coordinate planes.

1
b) Show that  (m, n) 
2
2
 sin
2 m 1
 cos 2 n 1  d .
0

c) Using gamma function evaluate
2

tan  d .
0
20. a) Solve p 2  q 2  z 2 ( x  y)
b) Solve the following by Charpit’s method
p 2  xp  q  0
c) Solve y 2 p  x 2 q  x 2 y 2 z 2
21. a) Verify Green’s theorem for  ( xy  y 2 )dx  x 2 dy where C is the region bounded by y=x and
c
y=x2.
b) Show that 18!+1 is divisible by 437.
22. a) State and prove Wilson’s theorem.
d2y
dy
b) Solve 2  4  5 y  tet given y(0)  y1 (0)  0 using Laplace transforms.
dt
dt



2