# Loyola College B.Sc. Mathematics April 2009 Algebra, Calculus & Vector Analysis Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE Examination – Mathematics

Third Semester – OCT/NOV 2010

MT 3501/MT 3500 – Algebra, Calculus and Vector Analysis

Date & Time:                                Dept. No.                                                    Max. : 100 Marks

PART – A

Answer ALL questions.                                                                                                  (10 ´ 2 = 20)

1. Evaluate
2. Find the Jacobian of the transformation x = u (1 + v) ; y = v (1 + u).
3. Find the complete solution of z = xp + yq + p2 – q2.
4. Solve
5. For , find div at (1, -1, 1)
6. State Green’s theorem.
7. What is L(f¢¢ (t))?
8. Compute
9. Find the sum and number of all the divisors of 360.
10. Define Euler’s function f(n) for a positive integer n.

PART – B

Answer any FIVE questions                                                                                          (5 ´ 8 = 40)

1. Evaluate by changing the order of the integration.
2. Express in terms Gamma functions.
3. Solve z2( p2+q2 + 1 ) = b2
4. Solve p2 + q2 = z2(x + y).
5. Find
6. Find
7. Prove that
8. Show that 18! + 1 is divisible by 437

PART – C

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

1. (a) Evaluate  taken through the positive octant of the sphere x2 + y2 + z2 = a2.

(b)  Show that

1. (a) Solve (p2 + q2) y = qz.

(b)  Solve (x2 – y2)p + (y2 – zx)q = z2 – xy

1. (a) Verify Gauss divergence theorem for taken over the region bounded by the planes x = 0, x = a, y = 0 y = a, z = 0 and z = a.

(b)  State and prove Fermat’s theorem.

1. (a) Using Laplace transform solve  given that .

(b)  Show that if n is a prime and r < n, then (n – r)!  (r – 1)! + (-1)r – 1 º 0 mod n.

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