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  Page 1  of 3   Model Question Paper with effect from 2018-19 17MAT31 USN Third Semester B.E.Degree Examination Engineering Mathematics-III (Common to all Branches) Time: 3 Hrs Max.Marks: 100 Note: Answer any FIVE full questions, choosing at least ONE question from each module. Module-1 1.   (a) Find the Fourier series expansion of   ,  x f    if    .0,sin 0,0     xin x xin x f    Hence deduce that   42...7.515.313.11        (08 Marks)   (b) Obtain the Fourier series of    x x f      valid in the interval   ., l l     (06 Marks)   (c)  Find the half-range cosine series of     2 1   x x f    the interval .10    x   (06 Marks)   OR 2.   (a) A periodic function    x f     of period ‘6’ specified by the following table over the interval   6,0 :  x  0 1 2 3 4 5 6    x f    9 18 24 28 26 20 9 Obtain the Fourier series up to second harmonics.  (08 Marks) (b) Obtain the Fourier series of      x x x f        2  valid in the interval   .2,0      (06 Marks) (c)  Find the half-range sine series of    24,cos 40sin      x for  x x for  x x f    ( 06 Marks) Module-2 3.   (a) If    101,1  2  x for  x for  x x f   , find the infinite Fourier transform of  f(x)  and hence evaluate dx x x x x x 2cossincos 03      (08 Marks) (b) Find the Fourier sine transform of 2 11  x   ( 06 Marks) (c) Solve 1012  0,296  uuuuu  nnnn     , by using z-transforms. ( 06 Marks) www.android.previousquestionpapers.com | www.previousquestionpapers.com | www.ios.previousquestionpapers.comwww.android.universityupdates.in | www.universityupdates.in | www.ios.universityupdates.in  Page 2  of 3   OR 17MAT31  4.   (a) Find the Fourier sine transform of .  x e  Hence show that .0., 21sin 02      medx xmx x  m     (08 Marks) (b) Find the  z  -transform of   42cos       n  ( 06 Marks) (c) Find the inverse  z  -transform of      141218  2   z  z  z   ( 06 Marks)   Module-3 5.   (a) Define Karl Pearson’s coefficient of correlation. If    is the acute angle between the lines of regression, then show that       r r   y x y x 222 1tan       . Explain the significance when .1&0    r r    (08 Marks) (b) Fit a best fitting straight line bax y   for the following data:  (06Marks)    x   1 2 3 4 5  y  10 12 13 16 19 (c) Find the real root of the equation 7log2 10    x x , lying between 0.4&5.3  correct to three decimal  places, using regula-falsi method.  (06Marks)   OR 6.   (a) Ten students got the following marks in Mathematics a   nd Electronics in a class test: Calculate the coefficient of correlation.  (08 Marks)  (b) Fit a best fitting parabola cbxax y    2 for the following data:  (06 Marks) (c) Find a real root of the equation 01cos3    x x correct to four decimal places, using  Newton-Raphson method. ( 06 Marks)   Module-4 7.   (a) Find the values  y  at (i  ) x=110 & (ii) x=390  using the following table gives the distance  y  (in nautical miles) of the visible horizon for the given heights  x   (in feet) above the earth’s surface:  ( 08 Marks) Roll No. 1 2 3 4 5 6 7 8 9 10  Marks in  Mathematics 78 36 98 25 75 82 90 62 65 39  Marks in  Electronics 84 51 91 60 68 62 86 58 53 47  x   1 2 3 4 5 6 7 8 9  y  2 6 7 8 10 11 11 10 9  x 100 150 200 250 300 350 400  y 10.63 13.03 15.04 16.81 18.42 19.90 21.27 www.android.previousquestionpapers.com | www.previousquestionpapers.com | www.ios.previousquestionpapers.comwww.android.universityupdates.in | www.universityupdates.in | www.ios.universityupdates.in  Page 3  of 3   17MAT31  (b) Using Newton’s ge neral interpolation formula, construct an interpolating polynomial for the following data: ( 06Marks)  (c) Using Weddle’s rule, evaluate       d   20 cos , by dividing   2,0     into six equal parts. ( 06 Marks)   OR 8.   (a) Use an appropriate interpolation formula to find (i) 24  y  and (ii) 54  y , given ,612 20    y  ,539 30    y  ,446 40    y  .343 50    y  ( 08 Marks)  (b) Using Lagrange’s interpolation formula to fit a polynomial for the following data:  ( 06 Marks)     0 1 3 4   -12 0 6 12 (c) Evaluate  82 10 log  xdx  , u sing Simpson’s  rd rule taking 7 equidistant ordinates. ( 06 Marks)   Module-5 9. (a) Verify Green’s theorem in the plane for      c dy xdx y xy  22  where C   is the closed curve bounded  by 2 &  x y x y    ( 08 Marks)  (b) U sing Stoke’s theorem, evaluate    dS n F   s ˆ      where k  yz  j xz i y F   2 3   and S is the surface of the paraboloid 22 2  y x z     bounded by 2   z  . ( 06 Marks)  (c) Prove that geodesics of a plane are straight lines. ( 06 Marks)   OR 10.   (a) Using Gauss divergence theorem, evaluate   dS k  xy j zxi yz   s    ˆˆˆ  where S is the surface of the sphere 2222 a z  y x    in the first octant.  (08 Marks)   (b) Derive Euler’s equation in the standard form viz., 0   y f  dxd  y f     (06 Marks)  (c) Find the extremal of the functional    dx x y y y      022 cos4 ;        y y    00  (06 Marks)   *****  x -3 0 1 3  f(x) 2 1 0 -1 www.android.previousquestionpapers.com | www.previousquestionpapers.com | www.ios.previousquestionpapers.comwww.android.universityupdates.in | www.universityupdates.in | www.ios.universityupdates.in
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