Code: A-35/C-35/T-35 Subject: MATHEMATICS-II
Time: 3 Hours June 2006 Max. Marks: 100
NOTE: There are 9 Questions in all.
· Question 1 is compulsory and carries 20 marks. Answer to Q. 1. must be written in the space provided for it in the answer book supplied and nowhere else.
· Out of the remaining EIGHT Questions answer any FIVE Questions. Each question carries 16 marks.
· Any required data not explicitly given, may be suitably assumed and stated.
Q.1 Choose the correct or best alternative in the following: (2x10)
a. Which of the following is an entire function
(A)
(B)
(C)
(D)
b. Let 
. Then which of
the following statements is not correct.
(A)
f is
differentiable at z = 0. (B) f is differentiable at z 
0.
(C) f is not analytic at z = 0. (D)
f is not analytic at z 0.
c. The image of a square
under the transformation 
is
(A) a square (B) a circle
(C) the upper half plane (D) the right half plane
d. The value of 
is
(A) 
(B)
0
(C) 
(D)
e. The value of line
integral 
,
where C is the segment of the parabola 
from (0, 0) to (1, 1) is
(A)
(B)
0
(C) 
(D)
f. The directional
derivative of 
at
the point 
in
the direction of 
is
(A)
(B)
(C) 0 (D) 1
g. The unit normal
to the surface 
at
the point 
is
(A)
(B)
(C) 
(D)
None of the above
h. Which of the following probability mass functions can define a probability distribution
(A)
(B) 
(C) 
(D) 
i. The expected value of a random variable X is 3 and its variance is 2. Then the variance of 2X + 5 is
(A) 8 (B) 9
(C) 10 (D) 11
j. The equation 
is
(A) Elliptic (B) Parabolic
(C) Hyperbolic (D) None of the above
Answer any FIVE Questions out of EIGHT Questions.
Each question carries 16 marks.
Q.2 a. Compute
the limit 
. (8)
b. Show that
cosh 
,
where 
. (8)
Q.3 a. Evaluate
the integral 
,
by contour integration. (10)
b. Evaluate 
, where C is the
circle 
traversed
counter clockwise. (6)
Q.4 a. Show
that the function 
in polar coordinates (r, 
)
is harmonic.
Find the corresponding harmonic conjugate function and construct the analytic
function f (z) = u + iv such that f(1) = 1. (8)
b. Find the Fractional
Linear Transformation which maps the unit disc 
onto the right half plane 
. (8)
Q.5 a. A thin rectangular
homogenous thermally conducting plate lies in the xy-plane defined by 
. The edge y = 0
is held at the temperature x (x – 1), while the remaining edges are held at temperature
. The
other faces are insulated and no internal sources and sinks are present. Find
the steady state temperature inside the plate. (10)
b. It has been claimed that in 60% of all solar heat installations, the utility bill is reduced by at least one third. Accordingly what are the probabilities that the utility bill will be reduced by at least one-third in
(i) four of five installations,
(ii) at least four of five installations. (6)
Q.6 a. Let the probability density function of a random
variable x be given by 
Find the corresponding cumulative distribution function and determine the probabilities that the random variable x will take a value
(i) greater than 
(ii)
between 0 and 
. (8)
b. Consider the heat flow
in a thin rod of length l,
l. The ends
x = 0 and x = l are insulated. The rod was initially at temperature f (x) = x. By
the method of Separation of Variables, find the temperature distribution u
(x, t) in the rod, where u (x, t) is governed by the partial differential equation
. (8)
Q.7 a. If
the amount of a cosmic radiation to which a person is exposed while flying by
jet across India is a random variable having the normal distribution with 
mrem and 
mrem, find the
probability that the amount of cosmic radiation to which a person will be
exposed on such a flight is between 4.00 and 5.00 mrem.
Given that F(1.10) = 0.8643, F(-0.59) = 0.2776, where F is the distribution function of the standard normal distribution. (4)
b. Let the probabilities that there are 0, 1, 2 and 3 power failures in a certain city during the month of July be respectively 0.4, 0.3, 0.2 and 0.1. Find the mean and variance of the number of power failures during the month of July in the city. (4)
c. Show
that 
is
a conservative force field. Find its scalar potential and the work done in
moving an object in this field from (1, -2, 1) to (3, 1, 4). (8)
Q.8 a. Find
the work done in moving the particle in the force field 
along the space curve 
, 
from x = 0 to x =
2. (8)
b. Evaluate 
, where 
, S is the surface
of the plane x + y + z = 1 in the first octant and 
is unit outward normal to the
surface S. (8)
Q.9 a. Verify
the divergence theorem for 
on the surface S of the sphere 
. (8)
b. Verify Stokes theorem for
, on the
surface S of the sphere 
above the xy-plane. (8)