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 best alternative in the following: (2x10)
a. The partial differential equation of the transverse vibrations of a string is
(A)
(B)
(C)
(D)
b. The partial differential equation fxx +2fxy+4fyy=0 is classified as
(A) Parabolic (B) elliptic
(C) Hyperbolic (D) None of these
c. The value of 
, C : |z|=2 is
(A) 3 (B) 2
(C) 1 (D) 0
d. Residue of tan z at z=p/2 is
(A) –1 (B) 1
(C) 0 (D) 2
e. The Taylor series expansion of 
in
|z|<1 is
(A)
(B)
(C) 
(D) 
f. Let X be normal with mean 10 and variance 4, then P(X<11) is
(A)
(B)
(C) 
(D)
g. If 
is a valid
probability mass function of x, then the value of K is
(A)
p (B)
(C) 
(D)
2
h. If X is the random variable representing the outcome of the roll of an ideal die, then E(X) is
(A) 3 (B) 2.5
(C) 3.5 (D) 4
i. If S is a closed
surface enclosing a volume V and if 
then 
is equal to
(A)
(B)
3S
(C) 3V (D)
j. The unit normal at (2,–2,3) to the surface x2y+2xy=4 is
(A) 
(B)
(C) 
(D)
Answer any FIVE Questions out of EIGHT Questions.
Each question carries 16 marks.
Q.2 a. An elastic string of Length l which is fastened at its ends x=0 and x=L is released from its horizontal position (zero initial displacement) with initial velocity g(x) given as
Find the displacement of the string at any instant of time. (10)
b. Solve by the method of separation of variables
(6)
Q.3 a. The frequency distribution is given as
(5)
Calculate the standard deviation and the mean deviation about mean.
b. Suppose the life in hours of a certain kind of radio tube has p.d.f.
Find the distribution function. What is the probability that none of the 3 tubes in a given radio set will have to be replaced during the first 150 hours of operation? What is the probability that all three of the original tubes will be replaced during the first 150 hours? (6)
c. A variate X has p.d.f.
|
x |
–3 |
6 |
9 |
|
P(x) |
1/6 |
1/2 |
1/3 |
Find E(X), E(X2) and E(2x+1)2. (5)
Q.4 a. Fit a Poisson distribution to the following data which gives the number of calls per square for 400 squares. (8)
|
No. of calls per square (x) |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
|
No. of squares (f) |
103 |
143 |
98 |
42 |
8 |
4 |
2 |
0 |
0 |
0 |
0 |
It is given that e–1.32=0.2674.
b. Find the directional
derivative of 
,
where 
,
at the point (2,0,3) in the direction of the outward normal to the sphere x2+y2+z2=14
at (3,2,1). (8)
Q.5 a. A fluid motion is given by
.
Is the motion irrotational? If so, find the velocity potential. (8)
b. A vector field is given by
Evaluate the line integral 
where C is a circular path given by
x2+y2=a2. (8)
Q.6 a. Find 
and S is
the surface of the sphere having centre (3,–1,2) and radius 3. (8)
b. Evaluate 
where 
and C is
boundary of triangle with vertices (0,0,0), (1,0,0) and (1,1,0). (8)
Q.7 a. Show that the function z|z| is not analytic anywhere. (8)
b. Show that the function u(x,y)=4xy–3x+2 is harmonic. Construct the corresponding analytic function w=f(z) in terms of complex variable z. (8)
Q.8 a. Find the Taylor Series expansion of a
function of the complex variable 
about the point z=4. Find the region
of convergence. (8)
b. Evaluate
where 
(8)
Q.9 a. Using complex variable techniques evaluate the real integral
. (10)
b. Determine the poles and residue at each pole of function f(z)=cot z.. (6)