LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – MATHEMATICS
THIRD SEMESTER – APRIL 2008
MT 3500 - ALGEBRA, CALCULUS & VECTOR ANALYSIS
Date : 26-04-08
Time : 1:00 - 4:00
Dept. No.
XZ 7
Max. : 100 Marks
PART – A
Answer ALL questions:
1. Show that  (n+1) = n (n).
2. Show that  (m, n)   (n, m)
(10 x 2 = 20 marks)
3. Form the partial differential equation by eliminating the arbitrary function from z  f ( x 2  y 2 ) .
2 z
 xy
x 2
4. Solve:



5. Show that 3 y 4 z 2 i  4 x3 z 2 j  3x 2 y 2 k is solenoidal.

6. Show that curl r  0
7. Find  (eat ) .
8. Find  3
.
s2
9. Define Euler’s function.
10. Find the number of integer, less than 600 and prime to it.


PART – B
Answer any FIVE questions:
11. Show that  (m, n)   (m, n  1)   (m  1, n) .
(5 x 8 = 40 marks)
12. Show that  (1/ 2)  
z
z
13. Solve:  6 x  3 y;  3x  4 y
x
y
14. Find the general integral of (mz-ny)p+(nx-lz)q=ly-mx
15. Find the directional derivative of xyz-xy2z3 at(1,2,-1) in the direction



of i  j  3 k



16. If F  3xy i  y 2 j find



F , d r where C is the curve y=2x2 from (0,0) to (1,2).
c
17. Find  [ f (t )] if
f (t )  t 1  t  1 for t  o
18. With how many zeros does
end.
PART – C
Answer any TWO questions:
19. a) Evaluate

1
2 x
x2
(2 x 20 = 40 marks)
xy dx dy
0
b) Evaluate
 xy dx dy over the region in the positive octant for which
x  y  1.
1
20. a) Find the complete integral of p 2  qy  z  0 using charpits method.





b) If v  w  r where w is a constant vector and r is the position vector of a point show that curl
v  2w .
21. a) Verify Stoke’s theorem for
F  (2 x  y )i  yz 2 j  y 2 zk where S is the upper half of the sphere x 2  y 2  z 2  1 and C its
boundary.




1
1
b) Find (i)   s ( s 2  9)  and (ii)   s 2 ( s 2  9) 




22. a) Solve using Laplace transforms
d 2 y dy 2

 t  2t given that
dt 2 dt
dy
 2 at t = 0.
y  4 and
dt
b) Find the highest power of 11 in
.


2