VTU Computer Science (Semester 3)
Engineering Mathematics 3
June 2012
Total marks: --
Total time: --
(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) Obtain the Fourier series for the function \(f(x)=\begin{cases}1+\frac{2x}{\pi}& -\pi\le x\le0\\[2ex]1-\frac{2x}{\pi}&0\le x\le \pi\end{cases}\) and deduce \( \dfrac{1}{1^2}+\dfrac{1}{3^2}+\dfrac{1}{5^2}+.......=\dfrac{\pi}{8}\)
7 M
1 (b) Find the half range cosine series for the function f(x)=(x-1)2 in 0 < x < 1.
6 M
1 (c) Obtain the constant term and the coefficient of the first sine and cosine terms in the Fourier expansion of y as given below.
x 0 1 2 3 4 5
y 9 18 24 28 26 20
7 M

2 (a) Express the function \(f(x)=\begin{cases}1&|x|\le1\\[2ex]0&|x|>1\end{cases}\) as a Fourier integral and hence evaluate \(\displaystyle\int\limits^{\infty}_{0}\dfrac {\sin x}{x}dx \)
7 M
2 (b) Find the sine and cosine transform of f(x)=e-ax, a>0
6 M
2 (c) Find the inverse Fourier sine transform of \( \dfrac {e^{-as}}{s} \)
7 M

3 (a) A tightly stretched string with fixed end points x=0 and x=l is initially at rest in its equilibrium position. If it is vibrating giving to each of its points a velocity λx(l-x), find the displacement of the string at any distance x from one end and at any time t.
7 M
3 (b) Find the temperature in a thin metal bar of length l where both the ends are insulated and the initial temperature in bar is sinπx.
7 M
3 (c) Find the solution of Laplace equation, \( \dfrac {\partial^2u}{\partial x^2}+ \dfrac {\partial^2 u}{\partial y^2}=0 \), by the method of seperation of variables.
6 M

4 (a) Fit a parabola y=a+bx+cx2 to the following data:
x -3 -2 -1 0 1 2 3
y 4.63 2.11 0.67 0.09 0.63 2.15 4.58
7 M
4 (b) A fertilizer company produces two products Naphtha and Urea. The company gets a profit of Rs. 50 per unit products of naphtha and Rs. 60 per unit product of urea. The time requirements for each product and total time available in each plant are as follows:
Plant Hours required Available hours
Naphtha Urea
A 2 3 1500
B 3 2 1500

The The demand for product is limited to 400 units. Formulate the LPP and solve it graphically.
6 M
4 (c) Solve the following using Simplex method:
Minimize Z=z1+4x2
Subject to constraints -x1+2x2≤6; 5x1+4x2≤40; xj≥0
7 M

5 (a) Use Regula-falsi method to find a root of the equation 2x-log10x=7 which lies between 3.5 and 4.
6 M
5 (b) Solve the relaxation method.
10x-2y-2z=6; -x+10y-2z=7; -x-y+10z=8
7 M
5 (c) Use the power method to find the dominant eigen value and the corresponding eigen vector of the matrix \(A=\begin{bmatrix}2&-1&0\\-1&2&-1\\0&-1&2\end{bmatrix} \) with the initial eigen vector as \(\begin{bmatrix}1&1&1\end{bmatrix}^T\)
7 M

6 (a) The following data is on melting points of an alloy of lead and zine where t is the temperature in Celsius and P is the percentage of lead in the alloy, tabulated for P=40(10)90 (i.e. P from 40 to 90 at intervals of 10). Find the melting point of the alloy containing 86% of lead.
P 40 50 60 70 80 90
t 180 204 226 250 276 304

7 M
6 (b) Using Lagrange's formula, find the interpolation polynomial that approximates to the functions described by the following table:
x 0 1 2 5
f(x) 2 3 12 147

and hence find f(3)
7 M
6 (c) Evaluate \(\displaystyle\int\limits_0^5\dfrac{dx}{4x+5}\), by using Simpson's \(\frac{1}{3}^{rd}\) rule, taking 10 equal parts. Hence find log5.
6 M

7 (a) Solve the partial differential equation \(\dfrac{\partial^2U}{\partial x^2}+\dfrac{\partial^2U}{\partial y^2}=-10(x^2+y^2+10)\) over the square with side x=0, y=0, x=3, y=3 with u0 on the boundary and mesh length h=1.
1 M
7 (b) Solve the heat equation \(\dfrac{\partial U}{\partial t}=\dfrac{\partial^2U}{\partial x^2}\) subject to the conditions U(0,t)=u(1,t)=0 and \[ u(x,0)=\begin{cases}2x &\text{for }0\le x\le \frac{1}{2}\\ 2(1-x) &\text{for } \frac{1}{2} \le x\le 1 \end{cases}\]
Taking h=1/4 and according to Bender Schmidt equation.
6 M
7 (c) Evaluate the pivotal values of the equation un=16 uxx taking h=1 upto t=1.25. The boundary conditions are u(0,t)=u(5,t)=0, ut(x,0)=0 and u(x,0)=x2(5-x).
7 M

8 (a) If \[ U(z)=\dfrac {2z^2+5z+14}{(z-1)^4}\], evaluate u2 and u3
6 M
8 (b) Find the Z-transform of
i) sin(3n+5)
ii) \[\dfrac {1}{(n+1)!}\]
7 M
8 (c) Solve the yn+2+6yn+1+9yn=2n with y0=y1=0 using Z-transforms.
7 M

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