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 = ex  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
A12361040
B1016896
C1696149
       (b) Fit a second degree parabola to the following data:   10
 
x
y
1.01.1
1.51.3
2.01.6
2.52.0
3.02.7
3.53.4
4.04.1
 Or
  (a) Find the mean and variance of bionomial Distribution ? 10
  
  (b) Fit a Poission distribution to the following :   10
x
y
046
138
222
39
41

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