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)
- Find when u = x2 – y2; v = x2 + y2
- Find the complete integral of z = px + qy +p2q2
- Find grad f if f = xyz at (1, 1, 1)
- Evaluate divergence of the vector point function
- Find L[sin2 2t]
- Find the sum of all divisors of 360.
- Find the remainder when 21000 divisible by 17.
PART – B
Answer any FIVE questions. (8 ´ 5 = 40)
- Change the order of integration and evaluate
- Express in terms of Gamma functions and evaluate
- Solve p2 + pq = z2
- Solve xp + yq = x
- Show that the vectoris irrotational.
- Evaluate: (a) L[cos 4t sin 2t] (b) L[e-3t sin2t]
- Show that 18! + 1 is divisible by 437.
PART – C
Answer any TWO questions. (2 ´ 20 = 40)
- (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
- (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
- (a) Solve y¢¢ + 2y¢ – 3y = sin t given that y(0) = y¢(0) = 0
(b) State and prove the Weirstrass inequality.
- (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.