VTU Computer Science (Semester 3)
Engineering Mathematics 3
December 2013
Total marks: --
Total time: --
INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary


1 (a) Find the Fourier series expansion of the function f(x) = |x| in (-? ?), hence deduce that π28=n=11(2n1)2π28=n=11(2n1)2
6 M
1 (b) Obtain the half range cosin series for the function, f(x)=(x-1)2 in the interval 0 ? x ? 1 and hence show that π2=8{112+132+152+......}π2=8{112+132+152+......}
7 M
1 (c) Compute the constant term and first two harmonics of the Fourier series of f(x) given by.
x0π32π3π4π35π32πf(x)1.01.41.91.71.51.21.0
7 M

2 (a) Open the Fourier cosine transform of f(x)=11+x2
6 M
2 (b) Find the Fourier transform of f(x)={1x2for|x|10for|x|>1 and evaluate nxcosxsinxx3dx
7 M
2 (c) Find the inverse Fourier transform of s1+s2
7 M

3 (a) Obtain the various possible solutions of two dimensional Laplace's equation, uxx + uyy=0 by the method of separation of variables,
7 M
3 (b) Solve the one dimensional wave equation. \[ C^2 \dfrac {\partial^2 u}{\partial x}=\dfrac {\partial^2 u}{\partial t^2}, 0 < x0 is constant (iii) ut(x,0)=0
7 M
3 (c) Obtain the D' Almbert's solution of the wave equation un=C2uxx subject to the conditions u(x,0)=f(x) and ut(x,0)=0
6 M

4 (a) Find the best values of a, b, c, if the equation y=a+bx+cx2 is to fit most closely to the following observations.
x 1 2 3 4 5
y 10 12 13 16 19
7 M
4 (b) Solve the following by graphical method to maximize z=50x+60y subject to the constraints. 2x+3y? 1500, 3x+2y?1500, 0? x ? 400 and 0 ? y ? 400.
6 M
4 (c) By using Simplex method. Maximize P= 4x1-2x2-x3 subject to the constraints.
x1+x2+x3?3, 2x1+2x2+x3?4, x1-x2?0, x1 ? 0 and x2 ? 0
7 M

5 (a) Using Newton-Raphson method find a real root of x sin x + cos x =0 nearer to ? carryout three iterations upto 4-decimal places.
7 M
5 (b) Find the largest eigen value and the corresponding eigen vector of the matrix.
[210121012] BY using the power method by taking the initial vector as [1 1 1]T carryout 5-iterations.
7 M
5 (c) Solve the following system of equations by relaxation method:
12x+y+z=31, 2x+8y-z=24, 3x+4y+10z=58
6 M

6 (a) A survey conducted in a slum locality reveals the following information as classified below.
7 M
6 (b) Determine f(x) as a polynomials in x for the data given below by using the Newton's divided difference formula.
x 2 4 5 6 8 10
f(x) 10 96 196 350 868 1746
7 M
6 (c) Evaluate 10x1+x2dx by using Simpsons (13)rd rule by taking 6-equal strips and hence deduce an approximate value of log e2.
6 M

7 (a) Solve the wave equation, 2ut2=42u\parialx2 subjects to u(0, 1)=0, u(4, t)=0, ut(x,0)=0 and u(x,0)=x(4-x) by taking h=1, K=0.5 upto 4-steps.
7 M
7 (b) Solve numerically the equation ut=2ux2 subject to the conditions u(0,t)=0=u(1,t),t?0 and u(x,0)=sin ?x, 0 ? x ? 1. Carryout computations for two levels taking h=1/3 and k=1/36.
7 M
7 (c) Solve uxx+uyy=0 in the following square region with the boundary conditions as indicated in the fig. Q7(c)

6 M

8 (a) Find the z-transform of (i) sinh n? (ii) cosh n? (iii) n2
7 M
8 (b) Find the inverse z-transform of 2z3+3z(z+2)(z4)
6 M
8 (c) Solve the difference equation yn+2 +6yn+1+9yn=2n with y0=y1=0 using Z transforms.
7 M



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