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: (2x10)
a. Eliminating a and b from the 
the partial
differential equation is
(A)
(B) 
(C)
(D) 
b. Solution of 
is
(A)
(B) 
(C)
(D) 
c. Residue of 
at z = 0 is
(A)
1 (B)
-1
(C)
2 (D)
0
d. The function 
is not differentiable
if the value of z is equal to
(A)
–1 (B)
1
(C)
2 (D)
0
e. The value of integral 
is given by
(A) 
(B)
(C)
(D)
f. The angle between the tangent to the curve 
at the point t = 1 and
t = -1 is
(A)
(B)
(C)
(D)
g. If 
where u, v are scalar fields and 
is a vector field, the value of 
is equal to
(A) 6 (B) -8
(C) 8 (D) 0
h. The
work done in moving a particle in the force field 
along the curve 
from x = 0 to x = 2 is equal to
(A) 15 (B)
17
(C)
16 (D)
21
i. The value of k
for the probability density function of a variate X is equal to
|
X |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
|
P(X) |
k |
3k |
5k |
7k |
9k |
11k |
13k |
(A) 1/30 (B) 1/40
(C)
49 (D)
1/49
j. In a Poisson distribution if 2P(x=1)=P(x=2),
then the variance is equal to
(A)
4 (B)
0
(C)
-1 (D)
2
Answer any FIVE Questions out
of EIGHT Questions.
Each question carries 16
marks.
Q.2 a. A tightly stretched string with fixed end points x = 0 and x = l
is initially in a position given by 
. If it is released from rest from this position, find the
displacement y (x,t). (8)
b. Solve the
equation
with boundary conditions u(x,0)=3sin(nπx), u(0,t)=0 and
u(1,t) = 0, where 0<x<1, t>0. (8)
Q.3 a. X is a continuous random variable with
probability density function given by 
Find standard deviation and also the mean
deviation about the mean. (8)
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. (8)
Q.4 a. A transmission line 1000 km. long is initially under steady-state conditions with potential 1300 volts at the sending end (x=0) and 1200 volts at the receiving end (x=1000). The terminal end of the line is suddenly grounded, but the potential at the source is kept at 1300 volts. Assuming the inductance and leakance to be negligible, find the potential v(x,t). (8)
b.
If
r is the distance of a point (x,y,z) from the origin, prove that 
is the unit vector in the direction OZ. (8)
Q.5 a. If
where S is the surface of the paraboloid 2z =x2+y2
bounded by z = 2. Evaluate 
using Stoke’s theorem. (8)
b. Determine whether 
is a conservative
field? If so find the scalar potential. Also compute the work done in moving the particle from (0,1,-1) to
(π/2,-1,2). (8)
Q.6 a. Evaluate 
where S is the surface
of Ellipsoid ax2+by2+cz2=1.
(8)
b. The following data are the number of seeds
germinating out of 10 on damp filter paper for 80 sets of seeds. Fit a binomial distribution to these data: (8)
|
X |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
|
F |
0 |
20 |
28 |
12 |
8 |
6 |
0 |
0 |
0 |
0 |
0 |
Q.7 a. Show that the function 
is not analytic at the
origin even though CR equations are satisfied thereof. (8)
b. Find the
bilinear transformation which maps the points z = 1, i , -1 onto the points w = i
, 0, -i. Find the image of |z|<1. (8)
Q.8 a. Find Taylor’s expansion of 
(8)
b. Evaluate 
. (8)
Q.9 a. Using
complex integration, show that 
(8)
b. Find the image in the w-plane of the disk
|z-1| ≤ 1, under the mapping
w = 1/z. (8)