LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – MATHEMATICS
THIRD SEMESTER – November 2008
MT 3500 - ALGEBRA, CALCULUS & VECTOR ANALYSIS
Date : 06-11-08
Time : 9:00 - 12:00
Dept. No.
AB 07
Max. : 100 Marks
SECTION-A
Answer all questions:
1
1. Evaluate
1
1
  
0
(10 x 2=20)
0
xyz dxdydz .
0
( x, y)
.
 (r ,  )
3. Obtain the partial differential equation by eliminating a and b from z  ( x  a ) ( y  b)
2. If x  r cos  , y  r sin  ,
find
 2z
 sin x
x 2
5. If  = 3x 2 y  y 3 z 2 , find grad  at (1, -2, -1).
6. State Stoke’s theorem.
7. Find L (e2t Sin 3t ) .
4. Solve
 1 
8. Find L1 
.
n 
 ( s  a) 
9. Find the number and sum of all the divisors of 360.
10. Find the number of integers less that 720 and prime to it.
SECTION-B
Answer any five questions:
11. Evaluate
(5 x 8=40)
  x dydx where R is the region bounded by the curves
2
y  x and y  x 2 .
R
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
13. Solve z  px  qy  1  p  q .
2
2
14. Solve x 2 p  y 2 q  ( x  y ) z .
15. Show that 2 (r m )  m(m  1)r m2 .
16. (a) Find L(t et Cos t ) .
 Sin at 
(b) Find L 
.
 t 


1  2s
17. Find L1 
.
2
2 
 ( s  2) ( s  1) 
18. Show that (18) ! 1 is divisible by 437.
1
SECTION-C
Answer any two questions:
(2 x 20=40)
a
19. (a) Change the order of integration
 
0
(b) Evaluate
dxdydz
   ( x  y  z  1)
3
2 ax
x 2 dxdy and evaluate the integral.
0
taken over the volume bounded by the plane
x  0; y  0; z  0; x  y  z  1 .

(c) Evaluate  e x dx .
2
o
20. (a) Solve xzp  zyq  xy
(b) Find a complete integral of ( p 2  q 2 ) y  qz .



21. Verify Gauss divergence theorem for F  y i  x j  z 2 k for the cylinderical region S given by
x 2  y 2  z 2  a 2 ; z  o and z  h .
22. Solve
d2y
dy
 4  5 y  4e3t given y (o)  2; y1 (o)  7 .
2
dt
dt
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2