Code: A-35/C-35/T-35                                                                     Subject: 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 best alternative in the following:                                         (2x10)

a. The image of the point z=2+3i under the transformation is   

(A)  1+2i.                                            (B)  1+8i.

(C)   0.                                                (D)  1-8i.

b. Let (i) and (ii) denote the facts

(i) :   f is continuous at z = 0

(ii) :   f is differentiable at z =0.                                                                      

                                                                       Then for function  which is correct statement?

(A) Both (i) & (ii) are true.                  (B)  (i) is true, (ii) is false.

(C)  (i) is false, (ii) is true.                     (D)  Both (i) & (ii) are false.

             c.   The order of the pole of the function  at  is

(A) 2                                                  (B)  1

(C)  0                                                  (D)  4

             d.   The value of the integral  where C is circle  traversed clockwise, is

(A) (B)

(C)  0                                                 (D)  i         

             e.   The curl of the gradient of a scalar function U is 

(A) 1                                                  (B) 

(C)  (D)  0

             f.    The value of the integral , where C is the curve  from x = 3  to x = 24, is 

(A) 156                                              (B)  153

(C)  150                                              (D)  158

             g.   The tangent vector to the curve whose parametric equation is  at t=2 is given by 

(A) (B)

(C) (D)

             h.   The cumulative distribution function F of a continuous variate X is such that F(a) = 0.5 and F(b) = 0.7. Then value of  is given as

(A) 0                                                  (B) 0.5

(C) 0.2                                                (D) 0.7

             i.    A discrete random variate X has probability mass function f which is positive at  and is zero elsewhere.  If , then the value of  is

(A) 1                                                  (B) 0

(C) (D)

             j.    A room has three lamp sockets.  From a collection of 10 light bulbs of which only 6 are good, a person selects 3 at random and puts them in a socket.  What is the probability that room will have light?

(A)  (B)

(C) (D)

Answer any FIVE Questions out of EIGHT Questions.

Each question carries 16 marks.

Q.2     a.   The two equal sides of an isosceles triangle are of length a each, and the angle between them has a probability density function proportional to  in the range  and zero otherwise.  Find the mean value and variance of area of triangle.                                                                 (8)

             b.   Show that if X has Poisson distribution with mean 1, then its mean deviation about mean is .                                                                   (8)

Q.3     a.   A person plays m independent games.  The probability of his winning any game is  (a, b are positive numbers).  Show that probability that the person wins an odd number of games is    .      (8)

             b.   An infinitely long uniform plane plate of breadth is bounded by two parallel edges and an end right angles to them. This end is maintained at temperature for all points and the other edges at zero temperature.  Determine the temperature at any point of the plate in the steady state.                (8)

Q.4     a.   Using method of separation of variables solve .                              (8)

b.   Verify Stoke’s theorem for the function , where C is the curve of intersection of cone z =  by the plane z = 4 and S is surface of cone below z = 4.                               (8)

Q.5     a.   Verify the Green’s theorem for  and C is the square with vertices , , , .       (8)

     b.   Show that the vector field  is conservative.  Find its scalar potential and the work done by it in moving a particle from  to (2, 3, 4).                            (8)

  Q.6     a. Find a normal vector and the equation to tangent plane to surface  at point (3, 4, 5).                                                                       (6)

             b.   If is a constant vector and .  Show that curl .         (5)

c.   Find the values of  and  so that the surfaces

 and  intersect orthogonally at .                 (5)

Q.7     a.   Show that the function  is not regular at the origin, although   C-R equations are satisfied at this point.                                                   (8)

b.   If f(z) = u + iv is an analytic function of z and find f(z) subject to the condition .                                          (8)

Q.8     a.   Discuss the transformation  and show that it maps the circle , onto an ellipse.  In particular discuss the case when a = 1.                (8)

             b.   Obtain the first three terms of the Laurent series expansion of the function  about the point z = 0 valid in the region .         (8)

Q.9     a.   Evaluate the integral , by contour integration.                        (10)

b.   Evaluate  where C is unit circle described in the positive direction.                    (6)