Code: AE35/AC35/AT35                                                                 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. Eliminating a and b from , we obtain the partial differential equation

(A)                           (B) 

(C)                            (D) 

b.         Solution of  is

(A)                (B)

(C)  (D)

c.         Residue of  at z = 0 is

(A)  1                                                   (B)  –1

(C)  2                                                   (D)  0

d.         The function w = log z is analytic everywhere except when z is equal to

(A)  –1                                                 (B)  1

(C)  2                                                   (D)  0

e.         If f(z) is analytic in a simply connected domain D and C is any simple closed curve inside D, then the value of  is given by

(A)  1                                                   (B)  2

(C)  0                                                   (D)  3

f.          If   and 0 elsewhere, is a  p.d.f. then the value of  k is equal to

(A)  4                                                   (B)  2

(C)  3                                                   (D)  1

g.         If X is a binomial variate with p = 1/5, for the experiment of 50 trials, then the standard deviation is equal to

(A) 6                                                    (B)  –8

(C) 8                                                    (D)

h.         A unit normal to at (0,1,2) is equal to

(A)                                 (B) 

(C)                                    (D) 

i.          If  , then  value of  is equal to

(A)  2u                                                 (B)  –u

(C)  3u                                                 (D)  5u

j.          is independent of the path joining any two points, if it is

(A)  irrotational field                              (B)  solenoidal  field

(C)  rotational field                                (D)  vector field

Answer any FIVE Questions out of EIGHT Questions.

Each question carries 16 marks.

Q.2     a.   A string is stretched and fastened to two points l apart. Motion is started by displacing the string in the form from which it is released at time t = 0. Show that the displacement of any point at a distance x from one end at time t is given by .                (8)

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

Q.3     a.   X is a continuous random variable with probability density function given by

find k and mean value of X.                                                                                (8)

             b.   The probability that a man aged 60 will live to be 70 is 0.65. What is the probability that out of 10 men, now 60, at least 7 will live to be 70?            (8)

Q.4     a.   Solve the telephone equation  when

 assuming that  is large compared with unity.                      (8)

b. Show that the vector field defined by the vector function

is conservative.                                                           (8)

Q.5     a.   Evaluate (8)

     b.   If  show that vector  and  satisfy the wave equation (8)

  Q.6     a. Using the Green’s theorem, show that

                   where n is the unit outward normal vector to C.                                                  (8)

             b.   Use the Divergence theorem to evaluate  where  and S is the boundary of the region bounded by the paraboloid z =  and the plane z = 4y.         (8)

  Q.7     a.   Show that the function  is continuous at the point z = 0, but not differentiable at z = 0.                                                                 (8)

             b.   Show that the function is harmonic. Find its conjugate harmonic function u(x,y) and the corresponding analytic function f(z).      (8)

  Q.8     a.   Evaluate the integral

 where (8)

             b.   Show that the function  is analytic in the region .

                   Obtain the Laurent series expansion about z = 0 valid in the region.                     (8)

Q.9     a.   Using complex integration, compute .                                 (8)

b.   Show that  under the mapping w = 1/z, all circles and straight lines in the z-plane are transformed to circles and straight lines in the w-plane.           (8)