LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

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

SECOND SEMESTER – APRIL 2011

# MT 2501/MT 2500 – ALGEBRA, ANAL.GEO & CALCULUS – II

Date : 08-04-2011 Dept. No. Max. : 100 Marks

Time : 9:00 – 12:00

__PART – A__

**Answer ALL questions:** **(10 x 2 = 20)**

- Evaluate .
- Evaluate .
- Solve: .
- Solve: .
- Prove that the series is convergent.
- Test for convergency the series .
- Find the general term in the expansion of .
- Prove that the coefficient of in the expansion of is .
- Find the equation of the sphere which has its centre at the point and touches the

plane .

- Find the distance between the parallel planes and

__PART – B__

**Answer any FIVE questions:** **(5 x 8 = 40)**

- Prove that .
- If ( n being a positive integer), prove that .

Also evaluate and .

- Solve: .

- Solve
- Test for convergency and divergency the series
- Show that the sum of the series .
- Show that if

- Find the equation of the plane passing through the points

.

__PART – C__

**Answer any TWO questions:** **(2 x 20 = 40)**

- a) Evaluate (10 marks)
- b) Find the area and the perimeter of the cardiod . (10 marks)
- a) Solve: . (10 marks)
- b) Discuss the convergence of the series for

positive values of . (10 marks)

21.a) Show that the error in taking as an approximation to is

approximately equal to when is small. (10 marks)

- b) show that (10 marks)

- a) A sphere of constant radius passes through the origin and meets the axes in A, B, C.

Prove that the centroid of the triangle ABC lies on the sphere

(10 marks)

- b) Find the shortest distance between the lines

.

Also find the equation of the line of shortest distance. (10 marks)

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