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

Time: 3 Hours                                                                                                     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. Let .  Then which of the following statements is not correct

(A)  f (z) is differentiable at z = 0.        

                   (B)  f (z) is differentiable at .

(C) f (z) is not analytic at z = 0.

(D) f (z) is not analytic at any point .

b. Which of the following mapping is conformal at z = 0

(A) .                                       (B)  .

(C)  w = cos z.                                    (D) w = sin z.

             c.   The Taylor’s series about z = 2 of the function  converges in the region

(A) .                                         (B) .

(C) .                                    (D) .

             d.   If  then grad v equals

(A)  .                        (B)  .

(C)  + grad u.                            (D)     grad u.

             e.   The surface integral  where S is the surface of the sphere  equals

(A) 0.                                                 (B) .

(C)  .                                              (D) .

             f.    From an urn containing 4 white, 5 black and 6 blue balls, 5 balls are chosen at random with replacement.  The expected number of blue balls selected is

(A) 2.5.                                               (B) 2.

                   (C) 1.5.                                               (D) 1.

             g.   The mean and variance of a binomial probability distribution are 1 and  respectively, then the probability that random variable takes value 0 is

(A)  .                                             (B)  .

                   (C) .                                             (D) .

             h.   One dimensional heat equation is given by

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

PART I

Answer any THREE questions. Each question carries 14 marks.

Q.2     a.   Prove that  is harmonic.  Find a function v that is conjugate harmonic to u and hence the analytic function  with .                                          (7)       

             b.   If f(z) is a regular function of z, then prove that .                 (7)

Q.3  a.   Find the image of the strip  under the mapping .           (4)

b.   Find the image of the circle  under the mapping .       (4)

             c.   Find the linear fractional mapping that maps the points i, 1, 2 + i to 4i, 3-i,  respectively.              (6)

Q.4     a.   Show that  where  and  is irrotational.  Find f (r) if it is also solenoidal.                                                                 (8)

b.   The temperature at a point  in a space is given by  A fly located at the point (4, 4, 2) desires to fly in a direction that gets cooler fastest.  Find the direction in which it should fly.  Also find the rate decrease of temperature in the direction of flight.                      (6)

Q.5     a.   Evaluate the line integral  where C is a simple closed path enclosing origin in its interior.                                           (7)

             b.   Show that the following line integral is independent of path C from points  to  and hence evaluate the integral (7)

  Q.6     a. Obtain d’Alembert’s solution of the wave equation  with initial conditions u(x, 0) = f (x), .                                  (11)

b.   A string stretching to infinity in both direction is given the initial displacement  and released from rest.  Determine the subsequent motion using d’Alembert’s solution obtained in part (a) of the question.                                                                                                                         (3)

PART II

Answer any THREE questions. Each question carries 14 marks.

Q.7     a.   State Cauchy integral formula for derivatives of an analytic function.  If , where C is the circle  find  using Cauchy integral formula.                               (7)

             b.   Identify the singularities of the function .  Classify the singularities and find the residues for each of them.                                       (7)

Q.8     a.   State Green’s theorem and use this theorem to show that for a solution      w (x, y) of Laplace’s equation  in a region R with boundary curve C and outer unit normal vector ,

(7)

             b.   Verify Stoke’s theorem for , where S is the surface of upper half of the sphere  and C is the circular boundary on XOY-plane.                                  (7)

Q.9       a.   Suppose that two teams A and B are playing a series of games.  Team A has a probability p of winning a game against team B.  The first team to win three games is declared winner of the series.  Find the probability distribution of number of games played in the series for declaring a winner.          (7)

             b.   The probability density function of a random variable X equals  and = 0, otherwise.  Find c.  Also find the probability that X takes a value greater than its expected value.          (7)

Q.10           a.                                                        A ticket office can serve 4 customers per minute.  The average number of customers arriving to the ticket office for purchase of tickets is 120 per hour.  Assuming number of customers arriving to the ticket office follow Poisson distribution, find the probability that ticket office is continuously busy during first 30 minutes of opening.                                           (6)

             b.   Sick leaves time X used by employees of a company in one month is roughly normal with mean 1000 hrs and standard deviation 100 hrs.  How much time t should be budgeted for sick leaves during next month if t is to be exceeded with a probability of 16%?

                   Also find the probability that in the next year no more than one month will have sick leave time more than 1200 hrs. (You may use the following values of distribution function  of standard normal distribution )                           (8)

Q.11           a. Evaluate the integral  where a > 0, by using contour integration.                (9)

              b.  Let S be a closed surface of volume V, containing the point P in its interior and let N be the outer unit normal to the surface S at a general point.  Show that div.                                                  (5)