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. Residue of 
at the point 
is
(A) 1 (B) –1
(C) 0 (D) 1/2
b. Real part of 
at all 
is:
(A) 
(B)
(C) 
(D)
c. 
where c is
the left half of the unit circle traversed in the clockwise direction is
(A) –2i (B) i
(C) –i (D) 2i
d. The image of 
where w-plane
under the mapping 
is
(A) 2u–1 = 0 (B) 2u+1 = 0
(C) 2v+1 = 0 (D) 2v–1 = 0
e. If 
, grad 
at the point (1,
–2, –1) is
(A) 
(B)
(C) 
(D)
f. Curl 
is
(A) –i (B) j
(C) 
(D)
–2i
g. If A is such
that 
then
is
called
(A) Rotational (B) Irrotational
(C) Conservative (D) non Conservative
h. If 
then 
is_______;
where 
(A) 1 (B) ½
(C) 0 (D) ¼
i. Eliminating
function from 
we
obtain the partial differential equation
(A) 
(B) 
(C) 
(D)
j. If 
,
then value of
div F is
(A) 
(B)
(C) 
(D)
Answer any FIVE Questions out of EIGHT Questions.
Each question carries 16 marks.
Q.2 a. A
tightly stretched string of length l with fixed ends is initially in an
equilibrium position. It is set vibrating by giving each point a velocity 
. Find the Displacement 
(8)
b. An
infinitely long plane uniform plate is bounded by two parallel edges and an end
at right angles to them. The breadth of the plate is 
; This end is maintained
at a temperature 
at all points and other edges are
at zero temperature. Determine the temperature at any point of the plate in the
steady state. (8)
Q.3 a. A
random variable X have the density function 
. If 
and zero otherwise; Find
(i) its distribution function.
(ii)
probabilities 
and
. (8)
b. In a production of iron rods, let the diameter X be normally distributed
(i)
What percentage of defectives can we expect? If the tolerance limits are set
at 
in.
(ii) How should we set the tolerance limits to allow for 4% defectives?
Assume 
(8)
Q.4 a. A continuous
random variable X has a pdf 
find K. Also find mean and
variance of this random variable. (8)
b. Prove that 
where 
and 
(8)
Q.5 a. Find the
directional derivative of 
in the direction of a unit vector
which makes an angle of 
with x-axis. (8)
b. Use
the Divergence theorem to evaluate 
; where 
and S is the
boundary of the region bounded by the paraboloid 
and the plane z = 4y. (8)
Q.6 a. Show that 
is independent of the path of
integration from (1,1,2) to (2,3,4) and hence evaluate it.
(8)
b. Verify the Green’s
theorem for 
and
C is the square with vertices at (0,0), 
(8)
Q.7 a. Show that the function
(8)
satisfies
Cauchy-Riemann equations at z = 0, but 
does not exist.
b. Find
the image of the region 
under the mapping 
(8)
Q.8 a. Obtain the Taylor series expansion of
about
z = 0. Also find its radius of convergence. (8)
b. Evaluate
the integral 
;
a > 0 by using contour integration. (8)
Q.9 a. Using
the method of separation of variables, solve 
; where u(x,0)=6e–3x. (8)
b. Evaluate by using Stokes’s
theorem 
where
c is the boundary the rectangle 
. (8)