LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

THIRD SEMESTER – APRIL 2011

MT 3501/ MT 3500 – ALGEBRA, CALCULUS AND VECTOR ANALYSIS

 

 

 

Date : 12-04-2011              Dept. No.                                                    Max. : 100 Marks

Time : 1:00 – 4:00

 

PART – A

Answer ALL questions.                                                                                                 (10 ´ 2 = 20)

1. Evaluate
2. Find when u = x² – y²; v = x² + y²
3. Solve
4. Find the complete integral of z = px + qy +p²q²
5. Find grad f if f = xyz at (1, 1, 1)
6. Evaluate divergence of the vector point function
7. Find L[sin² 2t]
8. Find
9. Find the sum of all divisors of 360.
10. Find the remainder when 2¹⁰⁰⁰ divisible by 17.

 

PART – B

Answer any FIVE questions.                                                                                        (8 ´ 5 = 40)

11. Change the order of integration and evaluate
12. Express in terms of Gamma functions and evaluate
13. Solve p² + pq = z²
14. Solve xp + yq = x
15. Show that the vectoris irrotational.
16. Evaluate: (a) L[cos 4t sin 2t]                   (b) L[e^-3t sin²t]
17. Find
18. Show that 18! + 1 is divisible by 437.

 

PART – C

Answer any TWO questions.                                                                                       (2 ´ 20 = 40)

19. (a) Evaluate where the region V is bounded by x + y+ z = a (a > 0),
    x = 0; y = 0; z = 0.

(b)  Evaluate  where R is the region in the positive quadrant for which
x + y £ 1.

(c)  Show that

20. (a) Solve (x² + y² + yz)p + (x² + y² – xz)q = z(x+y)

(b)  Find the complete integral and singular integral of p³ + q³ = 8z

21. (a) Solve y¢¢ + 2y¢ – 3y = sin t given that y(0) = y¢(0) = 0

(b)  State and prove the Weirstrass inequality.

22. (a) State and prove Wilson’s theorem.

(b)  Verify Green’s theorem in the XY plane for where C is the closed curve in the region bounded by y = x; y = x².

 

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