Engineering Mathematics 3 : Question Paper Dec 2015 - Computer Science Engg. (Semester 3) | Visveswaraya Technological University (VTU)

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Engineering Mathematics 3 - Dec 2015

Computer Science Engg. (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) For the function: $$ f(x)= \left\{\begin{matrix} x \ \ \ in &0< x< \pi \\ x-2\pi n & \pi < x< 2\pi \end{matrix}\right. $$ Find the Fourier series expansion and hence deduce the result $ \dfrac {\pi}{4}= 1 - \dfrac {1}{3} + \dfrac {1}{5} - \cdots \ \cdots $(7 marks) 1 (b) Obtain the half range Fourier cosine series of the function f(x)=x(l-x) in 0 ≤ x ≤ l.(6 marks) 1 (c) Find the constant term and first harmonic term in the Fourier expansion of y from the following table:

x	0	1	2	3	4	5
y	9	18	24	28	26	20

(7 marks) 2 (a) Find the Fourier transform of the function: $\left\{\begin{matrix} 1 \text{ for}&|x|\lt a \\\\ 0 \text{ for}&|x| \gt a \end{matrix}\right.$ and hence evaluate: $ \int^{\infty}_0 \dfrac {\sin x}{x}dx $ (7 marks) 2 (b) Obtain the Fourier sine transform of f(x)=e^-|x| and hence evaluate $$ \int^{\infty}_0 \dfrac {x \sin mx}{1+x^2} dx, \ m>0 $$(7 marks) 2 (c) Solve the integral equation: $$ \int^{\infty}_0 f(x) \cos pxdx = \left\{\begin{matrix} 1-p, &0\le p\le l \\0, & p>1 \end{matrix}\right. $$ and hence deduce the value $$ \int^{\infty}_0 \dfrac {\sin^2 t}{t^2}dt. $$(7 marks) 3 (a) Obtain the various possible solutions of the two dimensional Laplace's equation u_xx+u_yy=0 by the method of separation of variables.(7 marks) 3 (b) A string is stretched and fastened to two points 'l' apart. Motion is started by displacing the string in the form $ y=a \sin \left ( \dfrac {\pi x}{l} \right ) $ from which it is released at time t=0. Show that the displacement of any point at a distance 'x' from one end at time 't' is given by $ y(x, t)=a \sin \left ( \dfrac {\pi x}{l} \right )\cos \left ( \dfrac {\pi ct}{l} \right ). $(6 marks) 3 (c) Obtain the D'Alembert's solution of the wave equation u_tt=c² u_xx subjected to the conditions u(x, 0)= f(x) and $ u(x, 0) = f(x) \text { and } \dfrac {\partial u}{\partial t}(x, 0)=a. $(7 marks) 4 (a) For the following data fit an exponential curve of the form y=a e^bx by the method of least squares:

x	5	6	7	8	9	10
y	133	55	23	7	2	2

(7 marks) 4 (b) Solve the following LPP graphically:
Minimize Z=20x + 10y
Subject to the constraints: x+2y≤40
3x+y≥30
4x+3y≥60
x≥0 and y≥0(6 marks) 4 (c) Using Simplex method, solve the following LPP:
Maximize: Z=2x+4y
Subject to the constraint
3x+y≤22
2x+3y≤24
x≥0 and y≥0.(7 marks) 5 (a) Using the Regula-Falsi method to find the fourth root of 12 correct to three decimal places.(7 marks) 5 (b) Apply Gauss-Seidel method to solve the following of equations correct to three decimal places:
6x+15y+2z=72
x+y+54z=110
27x+6y-z=8.5
(Carry out 3 iterations).(6 marks) 5 (c) Using Rayleigh power method, determine the largest Eigen value and the corresponding Eigen vector of the matrix A in six iterations. Choose [1 1 1]^T as the initial Eigen vector: $$ A=\begin{bmatrix} 2 &-1 &0 \\-1 &2 &-1 \\0 &-1 &2 \end{bmatrix} $$(7 marks) 6 (a) Using suitable interpolation formulae, find y(38) and y(85) for the following data:

x	40	50	60	70	80	90
y	184	204	226	250	276	304

(7 marks) 6 (b) If y(0)=-12, y(1)=0, y(3)=6 and y(4)=12, find the Lagrange's interpolation polynomial and estimate y at x=2.(6 marks) 6 (c) By applying Weddle's rule, evaluate: $ \int^1_0 \dfrac {xdx}{1+x^2} $ by considering seven ordinates. Hence find the value of log_e2.(7 marks) 7 (a) Using finite difference equation, solve $ \dfrac {\partial^2 u}{\partial t^2 } = 4 \dfrac {\partial ^2 u}{\partial x^2} $ subject to u(0, t) = u(4, t)=0 u_t(x, 0) and u(x, 0) = x(4-x) upto four time steps. Choose h=1 and k=0.5.(7 marks) 7 (b) Solve the equation u_t=u_{xx subject to the conditions u(0, t)=0, u(1, t)=0, u(x,0)= sin (π x) for 0≤t≤0.1 by taking h=0.2.}(6 marks) 7 (c) Solve the elliptic equation u_xx+y_yy=0 for the following square mesh with boundary values as shown. Find the first iterative values of u(i=1-9) to the nearest integer.

(7 marks) 8 (a) Find the z-transform of 2n+sin (nπ/4)+1.(7 marks) 8 (b) Obtain the inverse z-transform of $ \dfrac {2z^2 + 3x}{(z+2)(z-4)} $(6 marks) 8 (c) Using z-transform, solve the following difference equation:
u_n+2 + 2u_n+1 + u_n=n with n₀=u₁ =0.(7 marks)

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