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)
- Find the Jacobian of the transformation x = u (1 + v) ; y = v (1 + u).
- Find the complete solution of z = xp + yq + p2 – q2.
- For , find div at (1, -1, 1)
- State Green’s theorem.
- What is L(f¢¢ (t))?
- Find the sum and number of all the divisors of 360.
- Define Euler’s function f(n) for a positive integer n.
PART – B
Answer any FIVE questions (5 ´ 8 = 40)
- Evaluate by changing the order of the integration.
- Express in terms Gamma functions.
- Solve z2( p2+q2 + 1 ) = b2
- Solve p2 + q2 = z2(x + y).
- Prove that
- Show that 18! + 1 is divisible by 437
PART – C
Answer any THREE questions. (2 ´ 20 =40)
- (a) Evaluate taken through the positive octant of the sphere x2 + y2 + z2 = a2.
(b) Show that
- (a) Solve (p2 + q2) y = qz.
(b) Solve (x2 – y2)p + (y2 – zx)q = z2 – xy
- (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.
- (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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