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    Reg.NO..!..r......r..........o.o..........


    Name ! ....................................o


                                  I & II SemesterB.Tech.Engg.(New Scheme)
                                     I)egreeRegularExaminationoMay 2008
                                   .,K6 EN 101: ENGG.MATIfiMATICS-I
                                   I



    Time : 3 Hours                                                                          Max. Marks : 100


      I. a) Solve+ : sin(x+y)
                           OX


          b) Solve . y tanx -- sin2x
                  :l

          c) obtaint{Ug}
                     L t )
                  t-t{ , t*t ,}
          d) obtain
                      Ls'+s+lJ

          e) If z -.ax+bYf (ax - by), prove that

               .dz    d
               b . * u3:2abz
                dx    dy

                                                           Zv'z'- 3xy - 4x - 7 atthepoint
          0 Find anequationfor thetangentplaneto the surface
            (1,-1, 2)

          g) Find a Fourier seriesto represent in the interval(4,1)
                                             x2


          h) Expand(x) ::-x,
                  f                              if 0 < x .%

                                  -x-                <1
                                            |,'t %.x
               ashalf-rangesineseries.                                                             (8x5=40)


                                                                                                      P.T.O.
 .'..:
t t
 tvl
IY.l.     r4705                                                   -2-                       ililt
                                                                                           Nl ilIil |ililil|
                                                                                      I|||l||il]ilt lffit
                             d v ^ ,
        il. a, ) S odl v exxf * y : x ' y o J
                                    J                                                                 7

                                                                        d'y
           b) Using the methodof variationof parameters
                                                      solve                   + 4y -tanZx             8
                                                                        dx2
                                       OR
           a) Solve (*y3 + y) dx + 2 (rzyz+ x + y4) dy - 0                                            7
                                  d 2 v - x dd v
                                             x
           b) S o l v e T f i
                       x                           *y:logx                                            8

m. a)           If f (t) is a periodicfunctionwith periodT, thenprovethat
                                  T

                                  J;. r(t)dt
                L {r1ty}-0              -st                                                           7
                                      1-e

           b) Apply convolutiontheorem evaluate
                                      to


                L-,ig1) (rt + bt
                    + a'
                                                                                                      I
                          Ltrl                     )J
                                       OR

           ")   State proveconvolution
                    and              theorem.                                                         7

           b) Evaluate usingLaplace
                     by           transform
                    -,.                                 *r
                             ''
                           -.+                               sin mt
                (i) J t" sint dt                    (ii)J?"                                        3+5


NIV.a) State and prove Euler's theorem.                                                               6

                            "A/
           b) If u - x2 tan-t
                              v               - yztan-l
                                                 ')
                                                          that          =                             4
                                                    /r,show #            5#
           c) If z - f (t,y) and X : eu+ e-u and y : e-u - €*u, prove that
                0z _ _ - X _ 0z - i ^Az
                     Oz                                                                               -
                _                          "Cy                                                        5
                au At                 ax
                                       OR
     llillll||]rltil
Illllllllllil lllt lill                         -3-                          M 14705


     a) Prove(i) V x(A+B) : V xA* V rB                                               10
                            -
                  (ii)v)<(0A) (v 0 )xA+o
                                       (v,<A)

                          derivative 0 -- x2y * 4xz2at (1,-2,-l ) in the direction
     b) Find thedirectional         of       z
        2i-j-2k.                                                                     F

                                                                                     5

 V. a) Obtainthe Fourier seriesfor f (x) : sin mx in the range(-n, n ) wherem is
       neitherzeronor an integer.

     b) Giventhe following table,obtaintheFourierseries
                                                      neglecting
                                                               termshigherthan
        first harmonics.


             xo           00   600   r20"   I 800     240"   300"

             v        7.9      7.2   3.6    0.s       0.9    6.8                     7

                    OR
     a) DeriveEuler'sformulae.                                                       8
     b) Findthehalf-range    series f (x) : (x-1)2in 0 < x< 1
                        cosine    for                                                7

				
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