The Institution of Engineers (India) A.M.I.E.T.E ECE Question Paper

[ganeshbala]

MATHEMATICS-II QUESTION PAPER

Time: 3 Hours June 2006 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. Which of the following is an entire function

(A) (B)
(C) (D)

b. Let . Then which of the following statements is not correct.

(A) f is differentiable at z = 0. (B) f is differentiable at z 0.
(C) f is not analytic at z = 0. (D) f is not analytic at z 0.

c. The image of a square under the transformation is

(A) a square (B) a circle
(C) the upper half plane (D) the right half plane

d. The value of is

(A) (B) 0
(C) (D)

e. The value of line integral , where C is the segment of the parabola from (0, 0) to (1, 1) is

(A) (B) 0
(C) (D)
f. The directional derivative of at the point in the direction of is

(A) (B)
(C) 0 (D) 1

g. The unit normal to the surface at the point is

(A) (B)
(C) (D) None of the above


h. Which of the following probability mass functions can define a probability distribution

(A) (B)
(C) (D)

i. The expected value of a random variable X is 3 and its variance is 2. Then the variance of 2X + 5 is

(A) 8 (B) 9
(C) 10 (D) 11

j. The equation is

(A) Elliptic (B) Parabolic
(C) Hyperbolic (D) None of the above


Answer any FIVE Questions out of EIGHT Questions.
Each question carries 16 marks.

Q.2 a. Compute the limit . (8 )

b. Show that cosh , where . (8 )

Q.3 a. Evaluate the integral , by contour integration. (10)

b. Evaluate , where C is the circle traversed counter clockwise. (6)

Q.4 a. Show that the function in polar coordinates (r, ) is harmonic. Find the corresponding harmonic conjugate function and construct the analytic function f (z) = u + iv such that f(1) = 1. (8 )

b. Find the Fractional Linear Transformation which maps the unit disc onto the right half plane . (8 )

Q.5 a. A thin rectangular homogenous thermally conducting plate lies in the xy-plane defined by . The edge y = 0 is held at the temperature x (x – 1), while the remaining edges are held at temperature . The other faces are insulated and no internal sources and sinks are present. Find the steady state temperature inside the plate. (10)

b. It has been claimed that in 60% of all solar heat installations, the utility bill is reduced by at least one third. Accordingly what are the probabilities that the utility bill will be reduced by at least one-third in
(i) four of five installations,
(ii) at least four of five installations. (6)

Q.6 a. Let the probability density function of a random variable x be given by
Find the corresponding cumulative distribution function and determine the probabilities that the random variable x will take a value
(i) greater than (ii) between 0 and . (8 )

b. Consider the heat flow in a thin rod of length l, l. The ends x = 0 and x = l are insulated. The rod was initially at temperature f (x) = x. By the method of Separation of Variables, find the temperature distribution u (x, t) in the rod, where u (x, t) is governed by the partial differential equation . (8 )



Q.7 a. If the amount of a cosmic radiation to which a person is exposed while flying by jet across India is a random variable having the normal distribution with mrem and mrem, find the probability that the amount of cosmic radiation to which a person will be exposed on such a flight is between 4.00 and 5.00 mrem.
Given that F(1.10) = 0.8643, F(-0.59) = 0.2776, where F is the distribution function of the standard normal distribution. (4)

b. Let the probabilities that there are 0, 1, 2 and 3 power failures in a certain city during the month of July be respectively 0.4, 0.3, 0.2 and 0.1. Find the mean and variance of the number of power failures during the month of July in the city. (4)

c. Show that is a conservative force field. Find its scalar potential and the work done in moving an object in this field from (1, -2, 1) to (3, 1, 4). (8 )

Q.8 a. Find the work done in moving the particle in the force field along the space curve , from x = 0 to x = 2. (8 )

b. Evaluate , where , S is the surface of the plane x + y + z = 1 in the first octant and is unit outward normal to the surface S. (8 )

Q.9 a. Verify the divergence theorem for on the surface S of the sphere . (8 )

b. Verify Stokes theorem for , on the surface S of the sphere above the xy-plane. (8 )