Code: A-35/C-35/T-35 Subject: MATHEMATICS-II

Time: 3 Hours December 2004 Max. Marks: 100

NOTE: There are 11 Questions in all.

· Question 1 is compulsory and carries 16 marks. Answer to Q. 1. must be written in the space provided for it in the answer book supplied.

· Answer any THREE Questions each from Part I and Part II. Each of these questions carries 14 marks.

· Any required data not explicitly given, may be suitably assumed and stated.

Q.1 Choose the correct or best alternative in the following: (2x8)

a. If is an analytic function of , then equals

(A) (B)

(C) (D)

b. The image of the circle under the mapping w = z +(3+2 i) is a

(A) circle. (B) ellipse.

(C) pair of lines. (D) hyperbola.

c. The function at z = 0 has

(A) a removable singularity.

(B) a simple pole.

(C) an essential singularity.

(D) a multiple pole.

d. A unit normal vector to the surface , at the point (1, 1, 1) is

(A) . (B) .

(C) . (D) .

e. The line integral , where C is the boundary of the region equals

(A) 0. (B) a.

(C) (D)

f. The partial differential equation is elliptic if

(A) (B)

(C) (D)

g. The expected value of a random variable X is 2 and its variance is 1, then variance of 3X+4 is

(A) 9. (B) 7.

(C) 3. (D) 13.

h. Let X be a random variable having a normal distribution. If P (X < 0) = P(X > 2) = 0.4, then mean value of X equals

(A) 0. (B) 1. (C) 1.5 (D) 2.

PART I

Answer any THREE questions. Each question carries 14 marks.

Q.2 a. Determine the analytic function f (z) = u +i v, given that . (8)

b. If w = u + i v is an analytic function, then show that the family of curves u (x, y) = a, cut the family of curves v (x, y) = b orthogonally, a, b being parameters. (6)

Q.3 a. Find the image of infinite strip , under the mapping . (7)

b. Find the linear fractional transformation that maps the points i, -1, 1 of z-plane into the points 0, 1, of w-plane respectively. Where in w-plane is the interior of unit disc mapped by the fractional transformation obtained? (7)

Q.4 a. Show that is irrotational and hence find its scalar potential. (8)

b. Find the directional derivative of the scalar function at the point (2, -1, 1) in the direction of the normal to the surface at the point (-1, 2, 1). (6)

Q.5 a. Find the work done by a force by moving a particle once around the circle . (7)

b. Show that the vector field is conservative. Hence evaluate the line integral along a path joining the points (0, 0, 0) to (1, 1, 1) (7)

Q.6 A rod of length has its lateral surface insulated and is so thin that heat flow in the rod can be regarded as one dimensional. Initially the rod is at the temperature 100 throughout. At t=0 the temperature at the left end of the rod is suddenly reduced to 50 and maintained thereafter at this value, while the right end is maintained at 100. Let u (x, t) be the temperature at point x in the rod at any subsequent time t.

(i) Write down the appropriate partial differential equation for u (x, t), with initial and boundary conditions.

(ii) Solve the differential equation in (i) above using method of separation of variables and show that Where is the constant involved in the partial differential equation. (3+11)

PART II

Answer any THREE questions. Each question carries 14 marks.

Q.7 a. Evaluate the complex integral . Also find . (6)

b. Find the residues of at its isolated singularities, using Laurent’s series expansions. (8)

Q.8 a. Let u (x, y) be continuous with continuous first and second partial derivatives on a simple closed path C and throughout the interior D of C. Show that where is the directional derivative of u along the outer normal to the curve C. (6)

b. Verify Gauss divergence theorem for on the surface

S of the cuboid formed by the planes x = 0, x = a, y = 0, y = b, z = 0 and

z = c. (8)

Q.9 a. The probability of an airplane engine failure (independent of other engines) when the aircraft is in flight is (1-P). For a successful flight at least 50% of the airplane engines should remain operational. For which values of P would you prefer a four engine airplane to a two engine one? (7)

b. If the resistance X of certain wires in an electrical networks have a normal distribution with mean of 0.01 ohm and a standard deviation of 0.001 ohm, and specification requires that the wires should have resistance between 0.009 ohm and 0.011 ohms, then find the expected number of wires in a sample of 1000 that are within the specification. Also find the expected number among 1000 wires that cross the upper specification.

(You may use normal table values ). (7)

Q.10 Suppose that certain bolts have length where X is a random variable with probability distribution function.

(i) Determine C so that with probability , a bolt will have length between 400 – C and 400 + C.

(ii) Find the mean and variance of bolt length L. Also find mean and variance of (2 L+5). (4+10)

Q.11 a. Evaluate the integral dx, using contour integration. (7)

b. Prove that . (7)