ENGG. MATHS-III-TRANSFORMS AND PDE-PRACTICE QUESTIONS FOR FOURIER SERIES

                                            MATHEMATICS-PDE AND TRANSFORMS
                                                       MODEL QUESTION PAPER
                                                         FOURIER SERIES
Marks:100                                                                                                                           Time:3 hours
                                                                   PART-A:
                                          ANSWER ANY 8 OF THE FOLLOWING:(8X2=16):
01.State the conditions for the fourier series to exist.
02.Write down the Euler's formulae.
03.Find the coefficients an for f(x)=xsinx having a periodicity 2Π for 0 04.State Parseval's theorem.
05.Find the RMS value of any function in fourier series.
06.(a) If f(x) is an odd function in (-1,1), find the values of a0 and an.
     (b)In fourier series expansion of f(x)=|sinx| in (-Π, Π), find the value of bn.
07.Obtain the sine series for unity in (0,Π).
08.Write the fourier sine series of k in (0,Π).
09.Explain “Harmonic analysis”. Give its method and applications.
10.Obtain the first two coefficients in fourier series given the following data:
            x   0   1   2   3   4   5
            y   4   8  15  7   6   2


                                                          PART-B:
                              ANSWER ANY 12 OF THE FOLLOWING:(12X7=84):
11.Find the fourier series for f(x)= sqrt(1-cosx) where -Π <x<Π .
   
                        ∞
12.Show that:   Σ   x2 = Π2/3 + 4Σ (-1)n cosnx/n2 where -Π<x<Π.

13.Expand coshax as a fourier series in -Π<x< Π.
 
14.Find the fourier series of f(x) where
                     f(x)  =  0     for-2<x<-1.
                             = 1+x for -1<x<0.
                             = 1-x  for 1<x<2.
15. Find the fourier series for f(x)=x-x2 in -1<x<1 and using this series find the RMS value of f(x) in the interval .


16.(a). Prove the Parseval's identity for the fourier series.
     (b).The following table gives the variations of a periodic function over a period T; θ = 2Πx/T.
           Show that f(x) = 0.75 + 0.37 cosθ + 1.004 sinθ.
               x   0   T/6   T/3   T/2   2T/3   5T/6   T
               y 1.98 1.3  1.05  1.3  -0.88  -0.25  1.98


17.Prove that in the interval 0<x<l,
                   x = 1 /2 - 4l/ Π2 ( cos ( Πx/l) + 1/32 cos (3Πx/l) +......)
    and deduce that  1/14 + 1/34 + 1/54 +...........= Π4/96.

18.Find the expansion of period 2Π for the function y=f(x) which is defined in
(0,2Π) by means of the table of values given below. Find the series up to the third
harmonic.
                 x   0   Π/3   2Π/3   3Π/3   4Π/3   5Π/3   2Π
                 y    1  1.4     1.9      1.7      1.5      1.2      1
19.Expand xsinx and xcosx separately as a sine series in 0<x< Π.

20.Expand f(x) in a series of Sine provided the following data:
                      f(x) = sinx for 0<x<Π/4.
                            = cosx for Π/4 <x< Π/2.

21.Find the half range sine series for f(x)= x(Π-x) in (0,Π).
Deduce that 1/13 – 1/33 + 1/53 - ….........= Π3/32.

22.Expand f(x)= e-x as a fourier series in the interval 0<x<2Π and hence deduce the series ->cosech2Π.

23.(a) State complex and exponential form of fourier series.

      (b)Expand cosax as a fourier series in -Π<x<Π. Try this using the above concept.

24.Find the complex form of the fourier series of f(x) = eax in -Π<x<Π. Hence deduce the value of
           Π/(a sinh aΠ).

25.(a).Obtain the expansion of xsinx qs a cosine series in (0,Π).
     (b).Hence deduce 1 + 2/ (1X3) – 2/(3X5) + 2/(5X7) -....... .

                                  ALL THE BEST AND BEST OF EFFORTS!!!
BY,
RANGARAJAN T.R.