LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – MATHEMATICS
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)
- Show that G (n+1) = n G(n).
- Show that
- Form the partial differential equation by eliminating the arbitrary function from .
- Show that is solenoidal.
- Show that curl
- Find ë .
- Find ë .
- Define Euler’s function.
- Find the number of integer, less than 600 and prime to it.
PART – B
Answer any FIVE questions: (5 x 8 = 40 marks)
- Show that.
- Show that é
- Find the general integral of
- Find the directional derivative of xyz-xy2z3 at(1,2,-1) in the direction
- If find where C is the curve y=2x2 from (0,0) to (1,2).
- Find ë if
- With how many zeros does end.
PART – C
Answer any TWO questions: (2 x 20 = 40 marks)
- a) Evaluate
- b) Evaluate over the region in the positive octant for which .
- a) Find the complete integral of using charpits method.
- b) If where is a constant vector and is the position vector of a point show that curl .
- a) Verify Stoke’s theorem for
where S is the upper half of the sphere and C its boundary.
- b) Find (i) ëùand (ii) ëù
- a) Solve using Laplace transforms
and at t = 0.
- b) Find the highest power of 11 in .
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