# RGPV Question Papers

## BE Engineering Mathematics 2nd Sem June 2004

Note:    Attempt  Any five questions.

Answer to the same question should be given at one place.

All question carry equal marks.

1   (a) Find the Fourier series expansion of  f (x) when:

F  (x) =  -π < x < 0

X,            0< x <π

Hence deduce  1/i2  + 1/32  + 1/52 +………   = π2/8

(b)  Find the Fourier series of:

πx,    0 ≤ x ≤ 1

π (2-x),     1 ≤ x ≤ 2

2    Obtain a half range cosine series for:

Kx,   0 ≤  x  ≤1/2

F  (x)  =  k (l- x),  l/2  ≤x ≤

(b)   Find the Laplace transform of t2  sing at.S

3   (a)  Computed the inverse Laplace transform of log   s( s+ 1)/ (s2 + 4).

(b)  Using Laplace transform, solve:

D3y/dt3  +  2d2y/ dt2  –  dy/dt  -2y  =  0

Givan that

y = 1, dy/dt  =2,  d2y/ dt2  =  2  at  t =0

4   (a)  Solve:

X d2y/dx2  -2  (x + 1) dy/dx  y  =  (x -2)  y  =  (x – 2)  e2x

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

d2y/ dx2  +  4y  = 4 tan  2x

5   (a) Solve by series method, the differential equation:

(2x+x3)  d2y/ dx2  –  dy/ dx  – 6xy  =  0

(b) Determaine tha analytic function  f  (z)  =  u  +  iv  whose real part is  e2x   (x  cos  2y  sin  2y).

6  (a)  Find the gernel solution of:

(2x  –  2yz  – y2)  p  +   (xy  + zx)  q  =  (xy –  zx)

(b)  A tightly stretched string with fixed  end points  x  = 0 and x  =  l  is  initially ina position, given by  y    =y0 sin  (πx / l).  If it is released from rest from this position, find the displacement  y  (x,t).

7  (a) find  A when div grad rm =  Ar m – 2 where  r2  =x2  +  y2  +  z2.

(b) Evaluate   F. dr where  F = yi  +   zj  +xk  and C is the circle x2 + y2 =  1,  z  = 0

8  (a) Calculate the mean and variance of a binomial distribution.

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

X: 0  1  2  3  4

Y:  1  1.8  1.3  2.5  6.3