Code:
AE-35/AC-35/AT-35 Subject:
MATHEMATICS-II
Time: 3 Hours 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. If the ends x = 0 and x = L are insulated in one dimensional heat flow problems, then the boundary conditions are
(A)
at t=0.
(B) 
at t=0.
(C)
.
(D) 
.
b. If 
is
a solution of 
,
then the value of C is
(A) 1. (B) 2 .
(C) 
. (D)
.
c. The curves 
and 
are orthogonal if
(A) u and v are complex functions. (B) u + iv is an analytic function.
(C) u – v is analytic function. (D) u + v is an analytic function.
d. The value of 
along the line x
= y is
(A)
(B)
(C) 
(D)
e. The critical
points of the transformation 
are given as
(A)
(B)
(C) 
(D)
f. If the mean and variance of binomial variate are 12 and 4, then the probabilities of the distribution are given by the terms in the expansion of
(A)
(B)
(C) 
(D)
g. The 
of a random
variable X is 
,
then E(X) is given as
(A) 5 (B) 6
(C) 7 (D) 8
h. If 
is a conservative
force field, then the value of curl is
(A) 0 (B) 1
(C) 
(D)
i. If 
and 
then 
is equal to
(A)
5u (B)
5
(C) 5
(D)
5
j. If 
then 
, where S is the
surface of unit sphere is
(A) 
(B)
(C) 
(D)
Answer any FIVE Questions out of EIGHT Questions.
Each question carries 16 marks.
Q.2 a. Solve
the differential equation 
for the conduction of heat along a
rod without radiations, subject to the following conditions:
(i)
u is not
infinite for 
.
(ii)
for x = 0 and x =
L.
(iii)
, for t = 0 between
x = 0 and x = L. (10)
b. Solve
subject
to the boundary condition 
as 
. (6)
Q.3 a. From a bag containing a black and b white balls, n ball are drawn at random without replacement. Let X denote the number of black balls drawn. Find the probability mass function of random variable X and compute expectation of Y = 2 + 3 X. (5)
b. If the probability of a bad reaction from a certain injection is 0.001, determine the chance that out of 2000 individuals more than two will get a bad reaction. (5)
c. If X is a continuous random variable with p.d.f. given by 
Find the value of k and mean value of X. (6)
Q.4 a. Using
the method of separation of variables, solve 
, where 
. (8)
b. If the directional
derivative of 
at
the point 
has
maximum magnitude 15 in the direction parallel to the line 
, find the values of a, b
and c. (8)
Q.5 a. If
where 
, show that 
. (8)
b. Show that the integral
is
independent of the path joining the points (1,2) and (3,4). Hence evaluate the
integral. (8)
Q.6 a. Use Stoke’s theorem to evaluate 
, where 
and C is the
bounding curve of hemisphere 
oriented in the +ve direction. (8)
b. The vector field 
is defined over
the volume of the cuboid given by 
, 
. Evaluate the surface integral 
, where S is the
surface of the cuboid. (8)
Q.7 a. Find
the points where C-R equations are satisfied for the function 
. Where does 
exist? Where 
analytic? (8)
b. Find analytic function 
where 
.
(8)
Q.8 a. Find the images in w-plane of
(i)
The circle
with centre 
and
radius 
.
(ii)
The interior
of the circle in (i) in z-plane, under the mapping 
. (8)
b. Expand
in Laurents
series valid for 
. (8)
Q.9 a. Evaluate
. (10)
b. Use Cauchy Integral
formula to evaluate 
, where C is the circle 
, traversed
counter clock wise. (6)