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VTU Electronics and Communication Engineering (Semester 3)
Engineering Mathematics 3
December 2016
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

Solve any one questions Q.1(a,b) and Q,2(a,b,c)
1(a) Explain $$f(x)= x-x^2$$/ as a Fourier series in the interval (-π,
π).
8 M
1(b) Obtain the half-range cosine series for the function $$f(x) = x(l-x)$$/ in the intervl 0≤x≤l.
8 M

2(a) Obtain the Fourier series of $$f(x)=\frac{\pi -x}{2}$$/ in 0≤x≤2π. Hence deduce that $$\frac{\pi }{4} = 1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+.....$$/
6 M
2(b) Find the half-range sine series for the function $$f(x)=\left\{\begin{matrix} \frac{1}{4}-x\ \ \text{in} & 0< x< 2\pi .\\ x-\frac{3}{4}\ \ \text{in}& 1/2< x< 1 \end{matrix}\right.$$/
5 M
2(c) Compute the constant term and the coefficient of the 1st since and cosine terms in the Fourier series of y as given in the following table
 x: 0 1 2 3 4 5 y: 4 8 15 7 6 2
5 M

Solve any one questions Q.3(a,b,c) and Q,4(a,b,c)
3(a) If $$f(x)\left\{\begin{matrix} 1-x^2; &|x|< 1 \\ 0; & |x|\geq 1 \end{matrix}\right.$$/. Find the Fourier transform of f(x) and hence find the value of $$\int_{ 0}^{\infty }\frac{x\cos x-\sin x}{x^3}dx.$$/
6 M
3(b) Find the Fourier sine and cosine transform of $$f(x)=\left\{\begin{matrix} x, & 0< x< 2\\ 0, & \ \ {elsewhere} \end{matrix}\right.$$/
5 M
3(c) Solving using Z-transform$y_{n+2}-4y_{n}=0$ given that y0=0,
y1=2.
5 M

4(a) Obtain the inverse Fourier sine transform of $$F_s(\alpha )=\frac{e^{-a\alpha }}{\alpha },a> 0$$/.
6 M
4(b) Find the Z-transform of $$2n+\sin \left ( \frac{n\pi }{4} \right )+1.$$/.
5 M
4(c) if $$U(z)=\frac{z}{z^2+7z+10}$$/, find the inverse Z-transform.
5 M

Solve any one questions Q.5(a,b,c) and Q,6(a,b,c)
5(a) Obtain the coefficient of correlation for the following data:
 x: 10 14 18 22 26 30 y: 18 12 24 6 30 36
6 M
5(b) By the method of least square find the straight line that best fits the following data:
 x: 1 2 3 4 5 y: 14 27 40 55 68
5 M
5(c) Use Newton-Raphson method to find a root of the equation tanx-x=0 near x=4.5. Carry out two iterations.
5 M

6(a) Find the regression line of y on x for the following data:
4
 x: 1 3 4 6 8 9 11 14 y: 1 2 4 5 7 8 9
Estimate the value of y when x =10.
6 M
6(b) Fit a second degree parabola to following data:
 x: 0 1 2 3 4 y: 1 1.8 1.3 2.5 6.3
5 M
6(c) Solve xex-2=0 using Regula-Falsi method.
5 M

Solve any one questions Q.7(a,b,c) and Q,8(a,b,c)
7(a) From the data given in the following table. Find the number of students who obtained less than 70 marks.
 Marks: 0-19 20-39 40-59 60-79 80-99 Number of students: 41 62 65 50 17
6 M
7(b) Find the equation of the polynomial which passes through the points (4,
-43),
(7,
83),
(9,
327) and (12,
1053). Using Newton's divided diffference interpoltation.
5 M
7(c) Compute the value $$\int_{0.2}^{1.4}(\sin x-\log x+e^x)dx$$/ using Simpson's $\frac{3^{th}}{8}$ rule taking six parts.
5 M

8(a) Using Newton's backward interpolation formula find the interpolating polynomial for the function given by the following table:
 x: 10 11 12 13 f(x): 22 24 28 34
Hence fine f(12.5).
6 M
8(b) The following table gives the premium payable at ages in years completed. Interpolate the premium payable at aage 35 completed. Using Langrange's formula.
 Age completed: 25 30 40 60 Premium in Rs: 50 55 70 95
5 M
8(c) Evaluate $\int_{4}^{5.2}log_c \ \ X$ dx taking 6 equal strips by applying Waddles rule.
5 M

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