LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – MATHEMATICS

THIRD SEMESTER – APRIL 2008
MT 3501 – ALGEBRA, CALCULUS AND VECTOR ANALYSIS
Date : 260408 Dept. No. Max. : 100 Marks
Time : 1:00 – 4:00
SECTION – A
Answer ALL questions.: (10 x 2 = 20 marks)
 Evaluate .
 If find the Jacobian of x and y with respect to r and .
 Solve
 Find the complete solution of
.
 Find at (2,0,1) for .
 State Stoke’s theorem.
 Evaluate ë (Sinh at).
 Evaluate ë^{1.}
 Find the sum of all divisors of 360.
 Compute (720).
SECTION – B
Answer any FIVE questions. (5 x 8 = 40 marks)
 By the changing the order of integration evaluate
 Express interms of Gamma function and evaluate .
 Obtain the complete and singular solutions of .
 Solve.
 Find if
 Evaluate (i) ë (ii) ë
 Find ë^{1}
 Show that if x and y are both prime to the prime n, then x^{n1}y^{n1} is divisible by n. Deduce that x^{12}y^{12} is divisible by 1365.
SECTION – C
Answer any TWO questions. (2 x 20 = 40 marks)
 a) Evaluate over the tetrahedron bounded by the planes and the coordinate planes.
 b) Show that .
 c) Using gamma function evaluate.
 a) Solve
 b) Solve the following by Charpit’s method
 c) Solve
 a) Verify Green’s theorem for where C is the region bounded by y=x and y=x^{2.}
 b) Show that 18!+1 is divisible by 437.
 a) State and prove Wilson’s theorem.
 b) Solve given using Laplace
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