AMIETE – ET/CS/IT (NEW SCHEME) – Code:
AE56/AC56/AT56
Subject: ENGINEERING 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. If 
be an analytic function, 
is equal to
(A) +1 (B) -1
(C) +2 (D) -2
b. Under the transformation 
circle 
is transformed in 
-plane into a
(A) straight line (B) parabola
(C) ellipse (D) circle
c. Value of the
integral 
where C is 
is
(A) 
(B) 
(C) 
(D) 
d. The position
vector of a particle at time t is 
. If at t = 1, the
acceleration of the particle be perpendicular to its position vector, then a is
equal to
(A) 0 (B) 1
(C) 
(D)
e. If the surface 
be orthogonal to the
surface 
at the point 
then b is equal to
(A) 0 (B) 1
(C) 2 (D) 3
f. If 
and 
is a constant vector,
curl 
is equal to
(A)
(B) 
(C) (D) 
g.
is equal to
(A)
2 (B) 3
(C)
4 (D) 6
h. The
differential equation of all spheres whose centres lie on z-axis is
(A) 
(B) 
(C) 
(D) 
i. The probability that a leap year selected at
random will contain 53 Saturdays is
(A) 0 (B) 
(C) 
(D) 
j. The variance of a Binomial distribution is
(A) np (B) npq
(C) 
(D) 
Answer any FIVE Questions out
of EIGHT Questions.
Each question carries 16
marks.
Q.2 a. If f(z) is an analytic function with constant
modulus, show that f(z) is constant. (8)
b. Show that the transformation 
maps the families of
lines x=constant and y=constant into confocal hyperbolas and confocal ellipses
respectively.
(8)
Q.3 a. State
and prove Cauchy’s integral formula. (2+6)
b. Evaluate 
, where C is 
. (8)
Q.4 a. Find
the directional derivative of 
at the point (2, -1,
1) in the direction of the vector i+2j+2k. (8)
b. Show that
(8)
Q.5 a. Use Green’s theorem to evaluate 
where C is a plane triangle enclosed by the lines 
and 
. (8)
b. Use divergence
theorem to evaluate 
, where 
and S is the surface
of the sphere 
. (8)
Q.6 a. Estimate the values of f(22) and f(42) from
the following available data
|
x: |
20 |
25 |
30 |
35 |
40 |
45 |
|
f(x): |
354 |
332 |
291 |
260 |
231 |
204 |
(4+4)
b. Find an approximate value of 
by calculating upto 4
decimal places, by Simpson’s 
rule; 
, dividing the range
into 10 equal parts. (2+6)
Q.7 a. Solve 
(8)
b. Solve 
(8)
Q.8 a. A
and B throw alternately with a pair of dice.
The one who throws 9 first wins.
If A starts the game, compare their chances of winning. (8)
b. An insurance
company insured 2000 scooter drivers, 4000 car drivers and 6000 truck
drivers. The probability of accident is
0.01, 0.03 and 0.15 respectively. One of
the insured persons meets an accident.
What is the probability that he is a car driver? (8)
Q.9 a. In 800 families with 5 children each, how
many families would be expected to have
(i) 2 boys or 3 boys (ii) atleast one boy
(iii)
5 girls (iv) 5 boys (2+2+2+2)
b. For a
continuous probability distribution
Find 
and the mean and the variance of the distribution. (2+3+3)