AMIETE ET/CS/IT (OLD
SCHEME)
Code: AE35/AC35/AT35 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 the best alternative in the following: (2
10)
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) 2v+1=0 (B) 2v 1=0
(C) 2u+1=0 (D) 2u 1=0
d. The value of 
is equal to
(A) 1 (B) 1
(C) 2 (D) 0
e. The invariant points of the transformation 
is given by
(A) ± i (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) 2√2
h. 
is equal to
(A) 0 (B) 1
(C) 
(D) 
i. A unit normal to 
at (0,1,2) is equal to
(A) 
(B) 
(C) (D)
j. 
is independent of the path joining any
two points if it is
(A) irrotational field (B) solenoidal field
(C) rotational field (D) vector field
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 0
C and 100C 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. (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 car is not 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. Verify the Greens theorem for 
C is the boundary of
the square (0, 0), 
, 
, 
. (8)
b. Use
the Divergence theorem to evaluate 
taken over the sphere 
and l, m,
n are the direction cosines of the
external normal to the sphere. (8)
Q.7 a. Show
that the function 
is continuous at the
point z = 0, but not differentiable at z = 0. (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
(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
transform 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)