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)
 Evaluate
 Find the Jacobian of the transformation x = u (1 + v) ; y = v (1 + u).
 Find the complete solution of z = xp + yq + p^{2} – q^{2}.
 Solve
 For , find div at (1, 1, 1)
 State Green’s theorem.
 What is L(f¢¢ (t))?
 Compute
 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 z^{2}( p^{2}+q^{2} + 1 ) = b^{2}
 Solve p^{2} + q^{2} = z^{2}(x + y).
 Find
 Find
 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 x^{2} + y^{2} + z^{2} = a^{2}.
(b) Show that
 (a) Solve (p^{2} + q^{2}) y = qz.
(b) Solve (x^{2} – y^{2})p + (y^{2} – zx)q = z^{2} – 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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