RGPV MATHEMATICS -II PAPER

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BE-202

B.E. (First /Second Semester)

EXAMINATION, June-2010

(Common for all Branches )

ENGINERRING MATHEMATICS-II

(BE-202)

Time : Three Hours               Maximum Marks : 100        Minimum Marks : 35

   Note : Attempt all questions. All question carry equal marks,

           One full question should be solved at one place.

Q.1. (a) Prove that :         ∞                                                      10

             x² = π² / 3 + 4  ∑   (-1)ⁿ      Cos nx / n² , -π < x  < π

                                    n =1  

             and hence show that :

                                       ∑ 1/ n²   = π² / 6

      (b) Applying convolution theorem find the inverse transform of  S² / (S²+ a²)²      10 

Or

                                               

      (a) If f(x) = πx             ,       -2 < x   <   0   10                                                                                          =π(2-x)   ,            0 < x   <   π
         

                     Show that in the interval ( 0, 2 )
         
 
                                     

               f(x) = π / 2 -4 / π [Cos πx / 1²  + Cos 3πx / 3² + Cos 5πx / 5² +........]
   
  

 

      (b) Find the Laplace transform of :                               10
            
        

                                                  (i)   t² Cosat 
            
      
                                                                                                                           (ii)  (Cosat -  Cosbt) / t

 

 

 

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Q.2.    (a) Obtain the series solution of the equation   :                10 
                  
                   

4x d²y / d²x  + 2(1-x) dy / dx - y  = 0
        
           

(b)  Solve by the method of variation of parameters :       10
                        
                     

(D² + 1 ) y  = x sin x

Or 

        (a)  Prove that :                                                                             10

                       Pn(x) = ( 1 / 2ⁿ ∟n )  dⁿ / d xⁿ  (x²-1)ⁿ

        ^{(b)  Solve  :                                                                               10}
                       
                            d²y / d²x - 2 tan x dy / dx + 5 y = e^x  Sec x

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Q.3.    (a) Solve :                  10

                       ^{( x² - y² - z²) p + 2xy q  = 2xz}
  

           ^{(b)  Solve  :         10}
                                 
 

d^2z / dy^2 - d^2z  /  dx.dy = Sinx Cos2y

Or 

            (a)   Solve  :         10

z = px + qy +(1+ P^2 + Q^2)½

           (b) Using the method of separation of variables , solve   10

          du / dx  = 2 du / dt  + u , where  u(x, 0) = 6 e^ (-3x)

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 Q. 4     (a) If  ---->

                        R      = xi +yj +zk  , prove that      10

                   (i) div (r^n R)  = (n+3) r^n

                   (ii) Curl
(r^n R)  = 0

              (b) Using Divergence theorem to evaluate :   10

                      









∫s F .ds ,

                 Where F = x^3i + y63 j + z^3 k and S is the surface of the sphere x^2 +y^2 = a^2

 Or

             (a) Find the directional derivatives of f = xy^2 + yz^3 at the point (2,-1,1) in the direction of vector i +2j+2k.                       10 

             (b) Show that the vector field given by :     10

               F= (x^2-yz)i + 9y^2-zx) + (z^2-xy)k

              is irrotational and find the scalar potentioal.

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 Q.5   (a) Compute by Fisher's index formula the quality index from the data given below : 10

Artical	Price	Total Value	Price	Total Value
A	12	36	10	40
B	10	16	8	96
C	16	96	14	9

      
(b) Fit a second degree parabola to the following data:   10

 

x	y
1.0	1.1
1.5	1.3
2.0	1.6
2.5	2.0
3.0	2.7
3.5	3.4
4.0	4.1

 Or

  (a) Find the mean and variance of bionomial Distribution ? 10

  

  (b) Fit a Poission distribution to the following :   10

x	y
0	46
1	38
2	22
3	9
4	1