Flowchart: Alternate Process: DECEMBER 2008Code: 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.                   The solution of the partial differential equation  is

 

(A)                             

(B) 

(C)                             

(D) 

 

b.         When a vibrating string has an initial velocity, its initial conditions are

(A)                                    (B)

(C)                                   (D) None of these

 

c.         Image of   under the mapping w = 1/z is

(A)  2 Im w + 1 = 0                                          (B)  2 Im w – 1 = 0

(C)  2 Rl w + 1 = 0                                          (D)  2 Rl w – 1 = 0

 

d.         The value of  is equal to

 

(A)  –1                                                 (B)  1

(C)  2                                                   (D)  0

 

e.         The invariant points of the transformation  are given by

(A)                                                  (B) 

(C)  0                                                   (D) 

 

f.          In a Poisson Distribution if 2P(x = 1) = P(x=2), then the variance is

 

(A)  4                                                   (B)  2

(C)  3                                                   (D)  1

 

g.         If V(X)=2, then V(2X+3) is equal to

(A) 6                                                    (B)  –8

(C) 8                                                    (D)

 

h.         is equal to

(A)  0                                                   (B)  –1

(C)                                    (D) 

 

i.          If  at (1,–2,–1) is  equal to

 

(A)                          (B)   

(C)                            (D) 

 

j.          If  is such that then  is called

 

(A)  irrotational                         (B)  solenoidal 

(C)  rotational                                       (D)  none of these

 

 

Answer any FIVE Questions out of EIGHT Questions.

Each question carries 16 marks.

 

 

  Q.2     a.   A bar 100 cm. long with insulated sides has its ends kept at 00C and 1000C until steady state conditions prevail. The two ends are then suddenly insulated and kept so. Find the temperature distribution.     (8)

                

             b.   Solve by method of separation of variables .                       (8)

 

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

                   find the standard deviation and also the mean deviation about the mean for the random variable X.                                                                (8)

                         

             b.   A car hire firm has two cars which it hires out day by day. The number of demands for a car on each day is distributed as a Poisson.  Distribution with mean 1.5. Calculate the proportion of days on which no car is used and the proportion of days on which some demand is refused. (given that )                                                                  (8)

 

  Q.4     a.   A string is stretched between the fixed points (0,0) and (l,0) are released at rest from  the initial deflection is given by  find the deflection of the string at any time t.         (8)                   

b.      If r and  have their usual meanings and is a constant vector, prove that .    (8)

 

  Q.5     a.   Show that the vector field  is conservative.  Find its scalar potential and the work done in moving a particle from (-1,2,1) to (2,3,4).                                               (8)

       

             b.   Find the values of constants  and  so that the surfaces  intersect orthogonally at the point    (1,-1,2).                                                                    (8)

 

  Q.6     a.   The cylinder  intersects the sphere . Find the surface area of the portion of the sphere cut by the cylinder above the yz plane and within the cylinder.                                  (8)

 

             b.   Use the Divergence theorem to evaluate  taken over the surface of the sphere , where l, m, n are the direction cosines of the external normal to the sphere.                                                (8)

         

  Q.7     a.   Show that the function  is analytical everywhere except on the half line .                                                                    (8)

 

             b.   If u is a harmonic function, then show that is not a harmonic function, unless u is a constant.                                                             (8)    

 

  Q.8     a.   Evaluate the integral , traversal counter clockwise.                  (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.   Using complex integration, compute .                                                    (8)

 

             b.   Show that the bilinear transformation  transforms the real axis in the z plane onto a circle in the w plane. Find the center and radius of the circle in the w plane. Find the point  in the z plane which is  mapped onto the centre of the circle in the w plane.                              (8)