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)**

- Evaluate
- Find when
*u = x*^{2}– y^{2}; v = x^{2}+ y^{2} - Solve
- Find the complete integral of
*z = px + qy +p*^{2}q^{2} - Find
*grad**f*if*f**= xyz*at (1, 1, 1) - Evaluate divergence of the vector point function
- Find
*L[sin*^{2}2t] - Find
- Find the sum of all divisors of 360.
- Find the remainder when 2
^{1000}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
*p*^{2}+ pq = z^{2} - Solve
*xp + yq = x* - Show that the vectoris irrotational.
- Evaluate: (a)
*L[cos 4t sin 2t]*(b)*L[e*^{-3t}sin^{2}t] - Find
- 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
*(x*^{2}+ y^{2}+ yz)p + (x^{2}+ y^{2}– xz)q = z(x+y)

(b) Find the complete integral and singular integral of *p ^{3} + q^{3} = 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 = x ^{2}*.