Q.No:1 CSIR Jun-2015
The Laplace transform of \(6t^3 + 3 \sin 4t\) is
(1)
\(\frac{36}{s^{4}} + \frac{12}{s^{2} + 16}\)
(2)
\(\frac{36}{s^{4}} + \frac{12}{s^{2} - 16}\)
(3)
\(\frac{18}{s^{4}} + \frac{12}{s^{2} - 16}\)
(4)
\(\frac{36}{s^{3}} + \frac{12}{s^{2} + 16}\)
Check Answer
Option 1
Q.No:2 CSIR Dec-2016
The Laplace transform of
\[ f(t)=
\begin{cases}
\frac{t}{T}, & 0 < t T
\end{cases}
\]
is
(1)
\(-(1-e^{-sT})/s^2 T\)
(2)
\((1-e^{-sT})/s^2 T\)
(3)
\((1+e^{-sT})/s^2 T\)
(4)
\((1-e^{sT})/s^2 T\)
Check Answer
Option 2
Q.No:3 CSIR Dec-2017
Consider the differential equation \(\frac{dy}{dt}+ay=e^{-bt}\) with the initial condition \(y(0)=0\). Then the Laplace transform \(Y(s)\) of the solution \(y(t)\) is
(1)
\(\frac{1}{(s+a)(s+b)}\)
(2)
\(\frac{1}{b(s+a)}\)
(3)
\(\frac{1}{a(s+b)}\)
(4)
\(\frac{e^{-a}-e^{-b}}{b-a}\)
Check Answer
Option 1
Q.No:4 CSIR Assam Dec-2019
The Laplace transform of the function \(y(t)\) which satisfies the differential equation \(\frac{dy}{dt}+y=te^{-2t}\), with the boundary condition \(y(0)=9\), is
(1)
\(\frac{9}{(s+1)(s+2)^2}+\frac{1}{s+1}\)
(2)
\(\frac{9}{s+1}+\frac{1}{(s+1)(s+2)^2}\)
(3)
\(\frac{9}{s+1}+\frac{1}{(s+2)^2}\)
(4)
\(\frac{9}{(s+2)^2}+\frac{1}{s+1}\)
Check Answer
Option 2
Q.No:5 CSIR Sep-2022
The Laplace transform \(L[f](y)\) of a function
\[ f(x)=
\begin{cases}
1, & 2n\leq x \leq 2n+1 \\
0, & 2n+1\leq x \leq 2n+2
\end{cases}
\]
n=0,1,2... is
(1)
\(\frac{e^{-y}(e^{-y}+1)}{y(e^{-2y}+1)}\)
(2)
\(\frac{e^{y}-e^{-y}}{y}\)
(3)
\(\frac{e^{y}+e^{-y}}{y}\)
(4)
\(\frac{e^{y}(e^{y}-1)}{y(e^{2y}-1)}\)
Check Answer
Option 4
Q.No:6 CSIR June-2023
The value of the integral \(\int_{-\infty}^\infty dx 2^{-\frac{|x|}{\pi}}\delta(\sin x)\) where \(\delta(x)\) is the Dirac delta function, is
1) \(3\)
2) \(0\)
3) \(5\)
4) \(1\)
Check Answer
Option 1
Q.No:1 GATE-2021
A function \(f(t)\) is defined only for \(t\geq 0\). The Laplace transform of \(f(t)\) is
\[
\mathcal{L}(f; s)=\int_{0}^{\infty} e^{-st} f(t)dt
\]
whereas the Fourier transform of \(f(t)\) is
\[
\tilde{f}(\omega)=\int_{0}^{\infty} f(t)e^{-i\omega t}dt.
\]
The correct statement(s) is(are)
(A)
The variable \(s\) is always real.
(B)
The variable \(s\) can be complex.
(C)
\(\mathcal{L}(f; s)\) and \(\tilde{f}(\omega)\) can never be made connected.
(D)
\(\mathcal{L}(f; s)\) and \(\tilde(f)(\omega)\) can be made connected.
Check Answer
Option B-D
Q.No:1 JEST-2014
The Laplace transformation of \(e^{-2t}\sin{4t}\) is
(a)
\(\frac{4}{s^2+4s+25}\)
(b)
\(\frac{4s}{s^2+4s+20}\)
(c)
\(\frac{4}{s^2+4s+20}\)
(d)
\(\frac{4s}{2s^2+4s+20}\)
Check Answer
Option c
Q.No:2 JEST-2018
The Laplace transform of \((\sin{(at)}-at\cos{(at)})/(2a^3)\) is
(A)
\(\frac{2as}{(s^2+a^2)^2}\)
(B)
\(\frac{s^2-a^2}{(s^2+a^2)^2}\)
(C)
\(\frac{1}{(s+a)^2}\)
(D)
\(\frac{1}{(s^2+a^2)^2}\)