Q.No:1 JAM-2015
The Fourier series for an arbitrary periodic function with period \(2L\) is given by \(f(x)=\frac{a_0}{2}+\sum_{n=1}^\infty a_n \hspace{1mm} cos \frac{n\pi x}{L}+\sum_{n=1}^\infty b_n \hspace{1mm} sin \frac{n\pi x}{L} \) For the particular periodic function shown in the figure, the value of \(a_0\) is
(A)
\(0\)
(B)
\(0.5\)
(C)
\(1\)
(D)
\(2\)
Check Answer
Option C
Q.No:2 JAM-2016
Fourier series of a given function \(f(x)\) in the interval \(0\) to \(L\) is
\(f(x)= \frac{a_0}{2}\sum_{n=1}^\infty a_n \hspace{1mm} cos(\frac{2\pi n x}{L})+\sum_{n=1}^\infty b_n \hspace{1mm} sin(\frac{2\pi n x}{L})\)
If \(f(x)=x\) in the region \((0,\pi),b_2\)=____________.
Check Answer
Ans (-0.5)
Q.No:3 JAM-2017
For the Fourier series of the following function of period \(2\pi\)
\[
f(x)=
\left\{
\begin{array}{ll}
0 & -\pi< x<0 \\
1 & 0<x<\pi
\end{array}
\right.\]
the ratio (to the nearest integer) of the Fourier coefficients of the first and the third
harmonic is:
(A)
1
(B)
2
(C)
3
(D)
6
Check Answer
Option C
Q.No:4 JAM-2018
The function \(f(x)=\left\{
\begin{array}{ll}
x & , -\pi<x<0 \\
-x & , 0<x<\pi
\end{array}
\right.\)
\\
is expanded as a Fourier series of the form \(a_0+\sum_{n=1}^\infty a_n \hspace{1mm} cos(nx)+\sum_{n=1}^\infty b_n \hspace{1mm} sin(nx)\). Which of
the following is true?
(A)
\(a_0 \neq 0, b_n=0\)
(B)
\(a_0 \neq 0, b_n\neq 0\)
(C)
\(a_0 = 0, b_n=0\)
(D)
\(a_0 = 0, b_n\neq0\)
Check Answer
Option A
Q.No:5 JAM-2021
In the Fourier series expansion of two functions \(f_1(t)=4t^2+3\) and \(f_2(t)=6t^3+7t\) in the interval \(-\frac{T}{2}\) to \(+\frac{T}{2}\) the Fourier coefficients \(a_n\) and \(b_n\) (\(a_n\) and \(b_n\) are coefficients of \(cos(n\omega t)\) and \(sin(n\omega t)\), respectively) satisfy
(A)
\(a_n=0 \) and \(b_n\neq 0\) for \(f_1(t)\) ; \(a_n \neq 0\) and \(b_n=0\) for \(f_2(t)\)
(B)
\(a_n\neq 0 \) and \(b_n = 0\) for \(f_1(t)\) ; \(a_n = 0\) and \(b_n\neq 0\) for \(f_2(t)\)
(C)
\(a_n\neq 0 \) and \(b_n\neq 0\) for \(f_1(t)\) ; \(a_n = 0\) and \(b_n\neq 0\) for \(f_2(t)\)
(D)
\(a_n=0 \) and \(b_n\neq 0\) for \(f_1(t)\) ; \(a_n \neq 0\) and \(b_n\neq 0\) for \(f_2(t)\)
Check Answer
Option B
Q.No:6 JAM-2022
The function \(f(x)=e^{sinx}\) is expanded as a Taylor series in \(x\),around \(x=0\) in the form
\(f(x)=\sum_{n=0}^\infty a_n x^n\).The value of \(a_0 +a_1 +a_2\) is
(A)
0
(B)
\(\frac{3}{2}\)
(C)
\(\frac{5}{2}\)
(D)
5
Check Answer
Option C
Q.No:7 JAM-2023
A periodic function \(f(x)=x^2\) for \(-\pi < x < \pi\) is expanded in a Fourier series. Which of the following statement(s) is/are correct?
A) Coefficients of all the sine terms are zero
B) The first term in the series is \(\frac{\pi^2}{3}\)
C) The second term in the series is -4cos \(x\)
D) Coefficients of all the cosine terms are zero