LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

           B.Sc. DEGREE EXAMINATION – MATHEMATICS

XZ 7

THIRD SEMESTER – APRIL 2008

MT 3500 – ALGEBRA, CALCULUS & VECTOR ANALYSIS

 

 

 

Date : 26-04-08                  Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

PART – A

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

1. Show that G (n+1) = n G(n).
2. Show that
3. Form the partial differential equation by eliminating the arbitrary function from .
4. Solve:
5. Show that is solenoidal.
6. Show that curl
7. Find  ë .
8. Find ë .
9. Define Euler’s function.
10. Find the number of integer, less than 600 and prime to it.

 

PART – B

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

1. Show that.
2. Show that é
3. Solve:
4. Find the general integral of
5. Find the directional derivative of xyz-xy²z³ at(1,2,-1) in the direction

of

1. If find where C is the curve y=2x² from (0,0) to (1,2).
2. Find ë  if

for

1. With how many zeros does         end.

                                                            

PART – C

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

1. a) Evaluate

1. b) Evaluate over the region in the positive octant for which .

1. a) Find the complete integral of using charpits method.

1. b) If where is a constant vector and is the position vector of a point show that curl .

1. a) Verify Stoke’s theorem for

where S is the upper half of the sphere  and C its boundary.

1. b) Find (i) ë^ùand (ii) ë^ù

1. a) Solve using Laplace transforms

given that

and at t = 0.

1. b) Find the highest power of 11 in     .

 

 

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