Mechanical Engineering (Semester 3)
TOTAL MARKS: 100
TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.
(2) Attempt any four from the remaining questions.
(3) Assume data wherever required.
(4) Figures to the right indicate full marks.
1 (a) Obtain the Fourier series for the function$$ f(x)=\left\{\begin{matrix} -\pi x&; 0\le x \le 1 \\\pi (2-x) &;1\le x\le 2 \end{matrix}\right. $$ and deduce that $$ \dfrac {\pi^2}{8}=\sum^\infty_{n=1}\dfrac {1}{(2n-1)^2} $$(7 marks)
1 (b) Obtain the half range Fourier sine for the function. $$ f(x)=\begin{bmatrix}1/4-x & 0<x<1/2 \\x-3/ 4&1/2<x<1 \end{bmatrix} $$(7 marks)
1 (c) Compute the constant term and the first two harmonics in the Fourier series of f(x) given by the following table.
| x | 0 | 1 | 2 | 3 | 4 | 5 |
| f(x) | 4 | 8 | 15 | 7 | 6 | 2 |
(6 marks)
2 (a) Find the fourier transform of $$ f(x)= \left\{\begin{matrix} 1-x^2&for &|x|\le 1 \\0 &for &|x| >1 \end{matrix}\right. $$ and hence evaluate $$ \int^\infty_0 \left ( \dfrac {x \cos x - \sin x}{x^3} \right )\cos \dfrac {x}{2}dx $$(7 marks)
2 (b) Find the Fourier cosine transform of $$ f(x)=\dfrac {1}{1+x^2} $$(7 marks)
2 (c) Solve the integral equation $$ \int^{\infty}_0d(\theta)\cos \alpha \theta d\theta=\left\{\begin{matrix} 1-\alpha&;&0 \le \alpha \le 1 \\0 &; & a>1 \end{matrix}\right. \ hence \ evaluate \ \int^{\infty}_0 \dfrac {\sin^2 t}{t^2}dt. $$(6 marks)
3 (a) Solve two dimensional Laplace equation uxx+uyy=0, by the method of separation of variables.(7 marks)
3 (b) Solve the one dimensional heat equations $$ \dfrac {\partial u}{\partial t}=\dfrac {c^2\partial^2 u}{\partial x^2}, 0\ltx\lt\pi $$="" under="" the="" conditions:="" <br=""> (i) u(0,+)=0,u(?,t)=0
(ii) u(x,0)=u0 sinx where u0 = constant ? 0.</x<\pi>(7 marks)
3 (c) Obtain the D' Almbert's solution of one dimensional wave equation.(6 marks)
4 (a) Fit a curve of the form y=aebx to the following data:
| x: | 77 | 100 | 185 | 239 | 285 |
| y: | 2.4 | 3.4 | 7.0 | 11.1 | 19.6 |
(7 marks)
4 (b) Using graphical method solve the L.P.P minimize z=20x1=10x2 subject to the constraints
x1+2x2?40; 3x1+x2?0; 4x1+3x2? 60; x1?0; x2?0(6 marks)
4 (c) Solve the following L.P.P miximize z=2x1 + 3x2 + x3, subject to the constraints
x1+2x2+5x3?19, 3x1+x2+4x3?25, x1?0, x2?0, x3?0 using simplex method.(7 marks)
5 (a) Using the Regular - falsi method, find the root of the equation xex =cosx that lies between 0.4 and 0.6 Carry out four interations.(7 marks)
5 (b) Using relaxation method solve the equations.
10x-2y-3z=205; -2x+10y-2z=154; -2x-y+10z=120(7 marks)
5 (c) Using the Rayleigh's power method, find the dominant eigen value and the corresponding eigen vector of the matrix $$ A=\begin{bmatrix} 6&-2 &2 \\ -2&3 &-1 \\2 &-1 &3 \end{bmatrix} $$ starting with the initial vector [1, 1, 1]T(6 marks)
6 (a) From the following table, estimate the number of students who have obtained the marks between 40 and 45:
| Marks | 30-40 | 40-50 | 50-60 | 60-70 | 70-80 |
| Number of student | 31 | 42 | 51 | 35 | 31 |
(7 marks)
6 (b) Using Lagrange's formula, find the interpolating polynomial that approximate the function described by following table:
| x | 0 | 1 | 2 | 5 |
| f(x) | 2 | 3 | 12 | 147 |
Hence find f(3).(7 marks)
6 (c) A curve is drawn to pass through the points given by the following table:
| x | 1 | 1.5 | 2 | 2.5 | 3 | 3.5 | 4 |
| y | 2 | 2.4 | 2.7 | 2.8 | 3 | 2.6 | 2.1 |
Using Weddle's rule, estimate the area bonded bt the curve, the x-axis and the lines x=1, x=4.(6 marks)
7 (a) Solve the Laplace's equation uxx+uyy=0, given that;
(7 marks)
7 (b) $$ Solve \ \dfrac {\partial^2u}{\partial t^2}=4 \dfrac {\partial^2 u}{\partial x^2} $$ subject to u(0,t)=0; u(4,t)=0; u(x,0)=x (4-x). Take h=1, k=0.5(7 marks)
7 (c) Solve the equation $$ \dfrac {\partial u}{\partial t}=\dfrac {\partial^2 u}{\partial x^2} $$ subject to the conditions u(x,0)=sinx, 0?x?1; u(0, t)=u(1, t)=0 using Schmidt's method. Carry out computations for two levels, taking h-1/3, k=1/36.(6 marks)
8 (a) Find the Z-transform of : $$ i) \ (2n-1)^2 \\ ii) \ \cos \left (\dfrac {n\pi}{2}+\pi/4 \right ) $$(7 marks)
8 (b) Obtain the inverse Z-transform of $$ \dfrac {4x^2-2z}{z^3-5z^2+8z-4} $$(7 marks)
8 (c) Solve the difference equation yn+2 +6yn+1+9yn=2n with y0=y1=0 using Z transforms.(6 marks)
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