# Document 15510835 ```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
Time : 9.00 – 12.00
Dept. No.
Max. : 100 Marks
PART – A
(10 x 2 = 20)
3 2
1. Evaluate
  xy  x  y  dydx .
0 1
1
2. Evaluate
x
7
(1  x)8 dx .
0

3. Eliminate the arbitrary constants from z  x 2  a
4. Find the complete solution for
 y
2
 b .
p  q  1.
5. Find  , if   xyz  xy 2 z 3 .
6. Prove that div r  3 , where r is the position vector.
7. Find L(Sin2t).



  s  2 6  .


1
1
8. Find L 
9. Find the number and sum of all the divisors of 360.
10. State Fermat’s theorem.
PART – B
(5 x 8 = 40)
a 2a x
11. Change the order of integration and evaluate the integral
 
0
1
 x 1  x 
m
12. Express
n
p
xydxdy .
x2
a
1
dx in terms of Gamma functions and evaluate the integral
0
13. Solve x 2 p  y 2 q  z 2 .
14. Solve p 2  q 2  x  y.
 Sin2t 
. .
 t 
15. Find L 


7s 1
  s  1 s  2  s  3  .


16. Find L1 
17. Show that 2 r n  n(n  1)r n 2 .
18. Show that 132 n 1  92 n 1 is divisible by 22.
 x 1  x 
5
0
3 10
dx .
PART – C
19. (a) Evaluate
(2x 20 = 40)
  xydxdy taken over the positive quadrant of the circle x
(b) Prove that   m, n  
2
 y2  a2 .
mn
.
mn
20. (a) Solve  p 2  q 2  y  qz.
(b) Solve (y+z)p + (z+x)q = x+y.
21.(a) Verify Stoke’s theorem for A  (2 x  y )i  yz 2 j  y 2 zk 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
d2y
dy
 4  13 y  2et given y(0) = 0 , y1(0)= -1.
2
dt
dt
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