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 best alternative in the following:                                         (2x10)

a.       The image of the line under the mapping is a

(A)  Circle.                                          (B)  Parabola.

(C)    Hyperbola.                                  (D)  Ellipse.

b.      The values of z for which is real is

(A)    any multiple of .                         (B)  odd multiples of .

(C)  all multiples of .                       (D)  any real number.

c.   The value of where z = x i y is

(A)    2                                                  (B)  0

(C)  - 4i                                               (D)  i

d.   Let .  Then which of the following is true:

(A) is not continuous at z = 0

(B) is differentiable at z = 0

(C) is not continuous at z = 0 but differentiable at z = 0

(D) is continuous at z = 0 but not differentiable at z = 0

e.   The value of integral is

(A)     0                                                  (B)  1

(C)  1+2i                                             (D) f.    The angle between the surfaces and at the point (1, 1, 1) is

(A) (B) (C) (D) g.   A force field is said to be conservative if

(A)     Curl = 0                                   (B)  grad = 0

(C)  Div = 0                                    (D)  Curl (grad ) = 0

h.   The partial differential equation is an example of

(A)    Hyperbolic equation                     (B) Elliptic equation

(C) Parabolic equation                         (D) None of these

i.    The inequality between mean and variance of Binomial distribution which is true is

(A)   Mean < Variance                          (B) Mean = Variance

(C) Mean > Variance                          (D) Mean Variance = 1

j.    Let f (x) be a probability density function defined by , for and f (x) = 0 for x < 0, then the value of cumulative distribution function at x =2 is

(A) (B) (C) (D) Answer any FIVE Questions out of EIGHT Questions.

Each question carries 16 marks.

Q.2     a.   Show that the function is analytic everywhere except on the half line y =1 , x 0.                                                                       (8)

b.   If u is a harmonic function of two variables (x,y),  then show that is not a harmonic function, unless u is a constant.                                        (8)

Q.3     a.   Evaluate the integral from point 1-2i to point 2-i along the curve C, .                                                             (8)

b.   Find the residue of the function at z = .             (8)

Q.4     a.   Find all possible Taylor’s and Laurent series expansions of the function about the point z = 1.                                          (10)

b.   Evaluate the integral , using contour integration.                          (6)

Q.5     a.   Find the directional derivative of the scalar point function at the point (2, 2, 2) in the direction of the normal to the surface at the point .      (6)

b.   If and are constant vectors and show that and hence show that , where r = (10)

Q.6     a.   Find the value of the surface integral where S is the curved surface of the cylinder bounded by the planes x = 0, x = 2.                                        (8)

b.   The vector field is defined over the volume of the cuboid given by , . Evaluate the surface integral , where S is the surface of the cuboid.     (8)

Q.7     a.   A tightly stretched string with end points fixed at x = 0 and x = L, is initially at rest in equilibrium state.  If it is set vibrating by giving to each of its points a velocity find the displacement of the string at any point x from one end, at any point of time t.                                                      (12)

b.   Evaluate the integral , traversed anti-clockwise.       (4)

Q.8     a.   A continuous type random variable X has probability density f(x) which is proportional to x2 and X takes values in the interval [0, 2]. Find the distribution function of the random variable use this to find P (X >1.2) and conditional probability P(X > 1.2/ X>1).                                                      (8)

b.   A firm plans to bid Rs.300 per tonne for a contract to supply 1000 tonnes of a metal.  It has two competitors A and B and it assumes that the probability that A will bid less than 300/- per tonne is 0.3 and that B will bid less than Rs.300 per tonne is 0.7.  If the lowest bidder gets all the business and the firms bid independently, what is the expected value of business in rupees to the firm.                                                (8)

Q.9     a.   Suppose that on an average 1 house in 1000 houses gets fire in a year in a district.  If there are 2000 houses in that district find the probability that exactly 5 houses will have fire during the year. Also find approximate probability using Poisson distribution.   (8)

b.   Derive the mean and variance of binomial distribution.                                         (8)