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VTU Computer Science (Semester 3)
Engineering Mathematics 3
December 2014
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) Expand f$f(x)=\sqrt[]{1-cosx},\ 0 < x< \ 2\pi$ in a Fourier series.$\frac{1}{1.3} +\frac{1}{3.5}+\frac{1}{5.7}+...$
7 M
1(b) find the half - range since of f(x)=[e^{x}] in (0,1).
6 M
1(c) In a machine the displacement y of a given point is given for a certain angle x as follows:
 x 0 30 60 90 120 150 180 210 240 270 300 330 y 7.9 8 7.2 5.6 3.6 1.7 0.5 0.2 0.9 2.5 4.7 6.8

find the constant term and the first two harmonics in Fourier series expansion y
7 M

2(a) find Fourier transform of e-xand hence evaluate $\int_{0}^{\infty }\frac{cosxt}{1+t{2}}dt.$
7 M
2(b) Find Fourier sine transform of $f\times= \left\{\begin{matrix} x &0< x\leq 1 \\ 2-x, &1\leq x< 2. \\ 0, &x< 2 \end{matrix}\right.$
6 M
2(c) Solve the integral equation $\int_{0}^{\oe } f (x)cos\lambda xdx=e^{-\lambda }$
7 M

3(a) Find various possible solution of one -dimensional heat equation by separable variable method.
10 M
3(b) A rectangular plate with insulated is 10cm wide and so long compared to its width that it may be considered infinite in length without introducing an appreciable error. If the temperature of the short edge $y^{-0}$ is given by
$u=20x,0\leq x\leq 5 \\ =20(10-x),5\leq x\leq 10$
and the two long edge x-10 as well as the other short edge are kept at$0^{0}C$. Find the temperature u(x,y),
10 M

4(a) Fit a curve of the the form $y^{-}ae^{bx}$ to the data
 x 1 5 7 9 12 y 10 15 12 15 21
7 M
4(b) Use graphical method to solve the folling LPP:
Minimze$z -20_{x1}+30_{x2}$
subject to $x_{1}+3_{x2}\geq 5;\\2_{x1}+2_{x2}\geq 20;\\3_{x1}+2_{x2}\geq 24;\\x_{1,}x_{2\geq }0,$
6 M
4(c) Solve the following Lpp by using simplex method: subject to$Z=3_{x1}+2_{x2}+5_{x3}$ Subject to $X_{1} + 2_{2}+x_{3}\leq 430\\3_{x1}+2x_{3}\leq 460\\x_{1}+4x_{2}\leq 420\\x_{1}\geq 0,x_{2} \geq 0.$
7 M

5(a) Use the gauss -Seidal itertive to solve the system of linear equations.
27x+6y-z85;6x+15y+2z-72;x+y+54z= 110.carry out3 iterations by taking the intial approximation to the solution as (2,3,2),Consider four decimal places at each stage for each variable.
7 M
5(b) Using the Newton-Raphson method,Find the real root of the equation xsinx+cosx-0 near to $x^{-}\pi$,carryout four itcrations(x in radians).
6 M
5(c) find the largest eigen value and the corresponding eigen vector of the matrix$A-\begin{Bmatrix} 4&1 &-1 \\2 &3 &-1 \\-2 &1 &5 \end{Bmatrix}$ by power method.take$\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}$ as the initial vector.Perform 5 iterpolation.
7 M

6(a) Find f(0.1) by using netwton's forward interpolation formula and f(4.99)by using Newton's backward interpolation formula form the data
 x 0 1 2 3 4 5 f(x) -8 0 20 58 120 212
6 M
6(b) find the interpolating polynomial f(x) by using Newton's divided difference interpolation's form the data:
 x 0 1 2 3 4 5 f(x) 3 2 7 24 59 118
6 M
6(C) Evaluate $\int_{0}^{1.2}e^{x}$dx using weddle's rule.Taking six cqual sub intervals,compare the result with exact value.
7 M

7(a) solve $\frac{\partial^2 }{\partial x^2}+\frac{\partial^2 }{\partial y^2}=0$ in the following square mesh.Carry out two iterations.

7 M
7(b) Solve the Poisson's equation$\Delta ^{2}u=8x^{2}y^{2}$ for the square mesh given below with u=0 on the boundary and mesh length h-1.

6 M

8(a) Find the Z-transforms of i)$\left ( \dfrac{1}{2} \right )^{n}+\left ( \dfrac{1}{3} \right )^{n}$ ii) $3^{n}cos\frac{\pi n}{4}$
7 M
8(b) State and prove inital value theorem in Z-transform.
6 M
8(c) Solve the difference equation
u_{n+2}-2u_{n+1}+u_{n}=2^{n};u_{n}=2,u_{1}=1.\]
7 M

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