LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

**B.Sc., **DEGREE EXAMINATION –** MATHEMATICS**

THIRD SEMESTER – NOVEMBER 2012

# MT 3501/3500 – ALGEBRA, CALCULUS AND VECTOR ANALYSIS

Date : 02-11-2012 Dept. No. Max. : 100 Marks

Time : 9.00 – 12.00

__PART – A__

ANSWER ALL THE QUESTIONS: (10 x 2 = 20)

- Evaluate .
- Evaluate .
- Eliminate the arbitrary constants from .
- Find the complete solution for
- Find , if .
- Prove that div , where is the position vector.
- Find L(Sin2t).
- Find .
- Find the number and sum of all the divisors of 360.
- State Fermat’s theorem.

__PART – B__

ANSWER ANY FIVE QUESTIONS: (5 x 8 = 40)

- Change the order of integration and evaluate the integral .
- Express in terms of Gamma functions and evaluate the integral .
- Solve
- Solve
- Find .
- Find .
- Show that
- Show that is divisible by 22.

__PART – C__

__ __

ANSWER ANY TWO QUESTIONS (2x 20 = 40)

- (a) Evaluate taken over the positive quadrant of the circle .

(b) Prove that

- (a) Solve

(b) Solve (y+z)p + (z+x)q = x+y.

21.(a) Verify Stoke’s theorem for taken over the upper half surface of

the sphere x^{2}+y^{2} +z^{2} = 1, z 0 and the boundary curve C, the x^{2}+y^{2} = 1, z=0.

(b) State and prove Wilson’s Theorem.

- Using Laplace transform solve the equation given y(0) = 0 , y
^{1}(0)= -1.

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