LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – MATHEMATICS

THIRD SEMESTER – APRIL 2008
MT 3500 – ALGEBRA, CALCULUS & VECTOR ANALYSIS
Date : 260408 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 .
 Solve:
 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
 Solve:
 Find the general integral of
 Find the directional derivative of xyzxy^{2}z^{3} at(1,2,1) in the direction
of
 If find where C is the curve y=2x^{2} from (0,0) to (1,2).
 Find ë if
for
 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) ë^{ù}and (ii) ë^{ù}
 a) Solve using Laplace transforms
given that
and at t = 0.
 b) Find the highest power of 11 in .
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