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. A solution of
(y – z) p + (z – x) q = x – y is
(A) x2 + y2 + z2 = c (B) xyz = c
(C) xy/z = c (D) x2 + y2 – z2 = c
b. The harmonic conjugate of u = x3 – 3xy2 is
(A) y3 – 3xy2 (B) 3x2y – y3
(C) 3xy2 – y3 (D) 3xy2 – x3
c. The
value of the integral 
where c is 
is
(A) πi (B) 0
(C) 2πi (D) 4πi
d. The
invariant points of the bilinear transformation 
is
(A) 1 ±
2i (B) –1± 2i
(C) ± 2i (D)
invariant point does not exist.
e. If
with y (0) = 1. The
value of k1, k2, k3, k4 by taking h = 0.1
using IV order Runge Kutta method is
(A) 0.05, 0.066, 0.066, 0.0833 (B) 0.066, 0.066, 0.05, 0.08
(C) 0.1,
0.2, 0.2, 0.08 (D) 0.1, 0.08, 0.2, 0.2
f. The
value of the integral 
by using Simpson’s
3/8th rule is
(A) 0 (B) 1.286
(C) 1.357 (D) 1.5
g. If 
is ant vector field 
then
(A) is said to be a
solenoidal vector.
(B) 
is said to be a
solenoidal vector.
(C) is said to be
irrotational.
(D) None
of these.
h. If
is a pdf then the
value of k is equal to
(A)
2 (B) 1/9
(C) 1 (D) 1/18
i. If x is a binomial
variate with p = 0.2, for the experiment of 50 trials, the standard deviation
is equal to
(A) –8 (B) 8
(C) 0 (D) 1
j. Green’s
theorem states that 
is
(A) 
(B) 
(C) 
(D) 
Answer any FIVE
Questions out of EIGHT Questions.
Each question
carries 16 marks.
Q.2 a. Construct
the analytic function f (z) = u
+ iv given
. (8)
b. Determine the region of the w-plane into which the lines parallel
to x and y axis in the z – plane are mapped by the transformation w = z2. (8)
Q.3 a. State
and prove Cauchy’s integral formula in the form
. (8)
b. Obtain the Laurent’s series
expansion of 
in the region 
. (8)
Q.4 a. Find
the values of ‘a’ and ‘b’ such that the surfaces ax2 – byz = (a + 2) x
and 4x2y + z3 = 4 cut
orthogonally at (1, –1, 2) (8)
b. Show that 
. (8)
Q.5 a. Verify
Green’s theorem for 
where c is bounded by y = x and y = x2. (8)
b. Evaluate
by using stoke’s theorem 
where c is the
boundary of the rectangle 0 ≤ x ≤ π : 0 ≤ y ≤ 1, z
= 3. (8)
Q.6 a. From
the following table, estimate the number of students who have obtained marks
between 40 and 45. (8)
Marks 30 - 40 40 - 50 50 - 60 60 -70 70
- 80
No. of
students 31 42 51 35 31
b. Evaluate 
using (i) Simpson’s 1/3rd rule (ii) Simpson’s 3/8th
rule.
(8)
Q.7
a. Solve the differential equation
. (8)
b. Using
the method of separation of variables solve 
. (8)
Q.8 a. If A and B are any two events prove that P(A
B) = P(A) + P (B) – P(A
B) and hence prove that if A, B and C are any three events. (8)
b. A
bag contains 3 coins of which one is two headed and the other two are normal
& fair. A coin is selected at random and tossed 4 times in succession. If
all the four times it
appears to be head what is the probability that the two headed coin was
selected. (8)
Q.9 a. An airline knows that 13% of the people who
make reservations on a certain flight will not turn up. Consequently their
policy is to sell 12 tickets which can
accommodate only 10. What is
the probability that everyone who turns up on a
given day is accommodated? (8)
b. In
a test on 2000 electric bulbs, it was found that the life of a particular make
was normally distributed with an average life of 2040 hrs and S.D of 60 hrs.
Estimate the number of bulbs likely to burn for
(i) more
than 2150 hrs
(ii) less than 1950 hrs
(iii) more than 1920 hrs
& less than 2160 hrs.
Assume A (1.83) = 0.4664, A (1.33)
= 0.4082, A (2) = 0.4772. (8)