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.
The
solution of the partial differential equation is
(A)
(B)
(C)
(D)
b. When a vibrating string has an initial velocity, its initial conditions are
(A) (B)
(C) (D)
None of these
c. Image of under the mapping w = 1/z is
(A) 2 Im w + 1 = 0 (B) 2 Im w – 1 = 0
(C) 2 Rl w + 1 = 0 (D) 2 Rl w – 1 = 0
d. The value of is equal to
(A) –1 (B) 1
(C) 2 (D) 0
e. The invariant
points of the transformation are given by
(A) (B)
(C) 0 (D)
f. In a Poisson Distribution if 2P(x = 1) = P(x=2), then the variance is
(A) 4 (B) 2
(C) 3 (D) 1
g. If V(X)=2, then V(2X+3) is equal to
(A) 6 (B) –8
(C) 8 (D)
h. is equal to
(A) 0 (B) –1
(C) (D)
i. If
at
(1,–2,–1) is equal to
(A) (B)
(C) (D)
j. If is such that
then
is called
(A) irrotational (B) solenoidal
(C) rotational (D) none of these
Answer any FIVE Questions out of EIGHT Questions.
Each question carries 16 marks.
Q.2 a. A bar 100 cm. long with insulated sides has its ends kept at 00C and 1000C until steady state conditions prevail. The two ends are then suddenly insulated and kept so. Find the temperature distribution. (8)
b. Solve
by method of separation of variables . (8)
Q.3 a. X
is a continuous random variable with probability density function given by
find the standard deviation and also the mean deviation about the mean for the random variable X. (8)
b. A car hire firm has
two cars which it hires out day by day. The number of demands for a car on each
day is distributed as a Poisson. Distribution with mean 1.5. Calculate the
proportion of days on which no car is used and the proportion of days on which
some demand is refused. (given that ) (8)
Q.4 a. A string is
stretched between the fixed points (0,0) and (l,0) are released at rest
from the initial deflection is given by find the deflection of the string
at any time t. (8)
b. If r and have their usual meanings
and
is
a constant vector, prove that
. (8)
Q.5 a. Show that the
vector field is
conservative. Find its scalar potential and the work done in moving a particle
from (-1,2,1) to (2,3,4). (8)
b. Find
the values of constants and
so that the surfaces
intersect orthogonally
at the point (1,-1,2). (8)
Q.6 a. The cylinder intersects the sphere
. Find the
surface area of the portion of the sphere cut by the cylinder above the yz
plane and within the cylinder. (8)
b. Use the Divergence
theorem to evaluate taken over the surface of the
sphere
,
where l, m, n are the direction cosines of the external normal to the sphere. (8)
Q.7 a. Show
that the function is analytical everywhere except on
the half line
. (8)
b. If
u is a harmonic function, then show that is not a harmonic function, unless u
is a constant. (8)
Q.8 a. Evaluate the integral , traversal counter clockwise.
(8)
b. Obtain
the first three terms of the Laurent series expansion of the function about the point
z = 0 valid in the region
. (8)
Q.9 a. Using
complex integration, compute . (8)
b. Show that the bilinear
transformation transforms the real axis in the z
plane onto a circle in the w plane. Find the center and radius of the circle in
the w plane. Find the point in the z plane which is mapped onto the centre of
the circle in the w plane. (8)