**LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034**

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

**SECOND SEMESTER – APRIL 2012**

**MT 2501/MT2500- ALGEBRA,ANALYTICAL GEOMETRY AND CALCULUS**

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**Date: 16-04-2012 Dept. No. Max. : 100 marks**

**Time: 9.00 – 12.00**

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__PART-A__

**ANSWER ALL QUESTIONS: **(10**×****2 =20) **

- Evaluate .
- Find x dx.
- Define exact differential equations.
- Solve (D
^{2}2D + 1) y = 0. - Show that the series is convergent.
- State Cauchy root test for convergence of a series.
- Find the coefficient of in the expansion of 1 + + + + …
- Prove that
- Find the direction cosines of the line joining the points (3,-5,4) and (1,-8,-2).
- Find the angle between the planes and .

__PART –B__

**ANSWER ANY FIVE QUESTIONS: (5****×****8=40)**

- Evaluate
- Find the surface area of the solid formed by revolving the cardiodabout the initial line.
- Solve
- Solve .
- Examine the convergence of
- Assuming that the square and the higher powers of x may be neglected

show that

- Sum to infinity the series
- Find the shortest distance between the linesand and the equation of the line.

__PART – C__

**ANSWER ANY TWO QUESTIONS: (2****×****20= 40)**

- (a) Evaluate.

(b) Find the length of the curve between the points given by and.

- (a) Solve

(b) Solve the following equation by the method of variation of parameter:

.

- (a) Test the convergence of the series

(b) Find the equation of the sphere passing through the points (2,3,1),(5,-1, 2),

(4,3,-1) and (2,5,3).

- (a) Show that

(b) Sum the series