LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034 LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034B.Sc. DEGREE EXAMINATION – MATHEMATICSTHIRD SEMESTER – November 2008MT 3501 – ALGEBRA, CALCULUS AND VECTOR ANALYSIS
Date : 06-11-08 Dept. No. Max. : 100 Marks Time : 9:00 – 12:00 PART – A (10 × 2 = 20 marks)
Answer ALL the questions
1. Evaluate .2. What is ?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 is zero.
6. State Gauss Divergence theorem.
7. Find L(cos23t).8. Find .9. Find Φ(360).
10. Find the highest power of 5 in 79!
PART – B (5 × 8 = 40 marks)
Answer any FIVE questions
11. Change the order of integration and evaluate .12. Prove that β(m,n+1 )+ β(m+1,n) = β(m,n).
13. Solve p tanx + q tany = tanz.
14. If are irrotational, prove that
(a) is solenoidal.
(b)Find the unit vector normal to the surface z = x2 + y2 – 3 at (2,-1,2). (4+4)15. Evaluate by Stokes Theorem where & C is the boundary of the triangle with vertices (0,0,0), (1,0,0) and (1,1,0).
16. Find (a) L(te-t sint).
(b)L(sin3t cosh2t).
17. Find .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 over the positive octant of the sphere x2+y2+z2 = a2
(b)Establish β(m,n) = . (10+10)
20. (a) Solve .
(b) Solve by Charpit’s Method, pxy + pq + qy = yz. (10+10)
21. (a) Verify Green’s theorem for where C is the boundary of the region x=0, y=0, x+y=1.
(b) Evaluate . (10+10)22. (a) Using Laplace Transform, solve given that y(0)=1, y`(0)=0..
(b) Using Wilson’s Theorem, prove that 10!+111 0 mod 143. (12+8)
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