STUPIDSID
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).
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.
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:
Estimate the value of y when x =10.
| x: | 1 | 3 | 4 | 6 | 8 | 9 | 11 | 14 |
| y: | 1 | 2 | 44 | 5 | 7 | 8 | 9 |
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.
-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:
Hence fine f(12.5).
| x: | 10 | 11 | 12 | 13 |
| f(x): | 22 | 24 | 28 | 34 |
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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