LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – MATHEMATICS
THIRD SEMESTER – November 2008
AB 08
MT 3501 - ALGEBRA, CALCULUS AND VECTOR ANALYSIS
Date : 06-11-08
Time : 9:00 - 12:00
Dept. No.
Max. : 100 Marks
PART – A
(10 × 2 = 20 marks)
Answer ALL the questions
1 2
  (x
1. Evaluate
2
 y 2 )dxdy .
0 0

2. What is
x e
2 x
dx ?
0
3. Find the complete integral of q = 2yp2
4. Write down the complete integral of z = px + qy + pq.
5. Find the constant k, so that the divergence of the vector F  ( x  z)i  (3x  ky) j  ( x  5z)k is zero.
6. State Gauss Divergence theorem.
7. Find L(cos23t).

8. Find  e 3t cos 3tdt .
0
9. Find Φ(360).
10. Find the highest power of 5 in 79!
PART - B
(5 × 8 = 40 marks)
Answer any FIVE questions
1 2 x
11. Change the order of integration and evaluate

xydydx .
0 x2
12. Prove that β(m,n+1 )+ β(m+1,n) = β(m,n).
13. Solve p tanx + q tany = tanz.
14. If A & B are irrotational, prove that
(a) A  B is solenoidal.
(b)Find the unit vector normal to the surface z = x2 + y2 – 3 at (2,-1,2).
(4+4)
1
15. Evaluate
 F .d r
by Stokes Theorem where F  y 2 i  x 2 j  ( x  z )k & C is the boundary of the
C
triangle with vertices (0,0,0), (1,0,0) and (1,1,0).
16. Find (a) L(te-t sint).
(b)L(sin3t cosh2t).
 s 2  3s  3 
17. Find L1 
.
2 
 ( s  3)( s  1) 
18. (a) If N is an integer, prove that N5-N is divisible by 30.
(6+2)
(b)State Fermat’s Theorem.
PART - C
(2 × 10 = 20 marks)
Answer any TWO questions
19. (a) Evaluate

dxdydz
a x y z
2
(b)Establish β(m,n) =
2
2
2
over the positive octant of the sphere x2+y2+z2 = a2
(m)(n)
.
 ( m  n)
(10+10)
20. (a) Solve x( z 2  y 2 ) p  y( x 2  z 2 )q  z ( y 2  x 2 ) .
(b) Solve by Charpit’s Method, pxy + pq + qy = yz.
(10+10)
21. (a) Verify Green’s theorem for  (3x 2  8 y 2 )dx  (4 y  6 xy )dy where C is the boundary of the
c
region x=0, y=0, x+y=1.


s
(b) Evaluate L1  2
.
2 2 
 (s  a ) 
(10+10)
d 2 y (t )
dy (t )
4
 3 y (t )  et given that y(0)=1, y`(0)=0..
22. (a) Using Laplace Transform, solve
2
dt
dt
(b) Using Wilson’s Theorem, prove that 10!+111  0 mod 143.
(12+8)
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