# Laplace Transform

 Question 1
Which of the following statements is true about the two sided Laplace transform?
 A It exists for every signal that may or may not have a Fourier transform. B It has no poles for any bounded signal that is non-zero only inside a finite time interval. C The number of finite poles and finite zeroes must be equal. D If a signal can be expressed as a weighted sum of shifted one sided exponentials, then its Laplace Transform will have no poles.
GATE EE 2020   Signals and Systems
Question 1 Explanation:
It has no poles for any bounded signal that is nonzero in a finite time interval. This is true as we know for finite amplitude finite width signal ROC is entire s plane and ROC never includes any pole.
It implies for such signals there is no poles. Hence the correct answer is option (B).
 Question 2
The output response of a system is denoted as y(t), and its Laplace transform is given by
$Y(s)=\frac{10}{s(s^2+s+100\sqrt{2})}$
The steady state value of y(t) is
 A $\frac{1}{10\sqrt{2}}$ B $10\sqrt{2}$ C $\frac{1}{100\sqrt{2}}$ D $100\sqrt{2}$
GATE EE 2019   Signals and Systems
Question 2 Explanation:
\begin{aligned} &=\lim_{s \to 0}sY(s)\\ &=\lim_{s \to 0}\frac{10s}{s(s^2+s+100\sqrt{2})}\\ &=\frac{10}{100\sqrt{2}}=\frac{1}{10\sqrt{2}} \end{aligned}
 Question 3
A system transfer function is $H(s)=\frac{a_1s^2+b_1s+c_1}{a_2s^2+b_2s+c_2}$. If $a_1=b_1=0$, and all other coefficients are positive, the transfer function represents a
 A low pass filter B high pass filter C band pass filter D notch filter
GATE EE 2019   Signals and Systems
Question 3 Explanation:
\begin{aligned} H(s)&=\frac{c_1}{a_2s^2+b_2s+c_2}\\ & as \;\; a_1=b_1=0\\ &=\frac{c_1}{(1+s\tau _1)(1+s\tau _2)}\\ \text{Put }s&=0,\; H(0)=\frac{c_1}{c_2}\\ \text{Put }s&=\infty ,\; H(\infty )=0 \end{aligned} which represents second order low pass filter.
 Question 4
The inverse Laplace transform of
$H(s)=\frac{s+3}{s^2+2s+1} \; for \; t\geq 0$ is
 A $3te^{-t}+e^{-t}$ B $3e^{-t}$ C $2te^{-t}+e^{-t}$ D $4te^{-t}+e^{-t}$
GATE EE 2019   Signals and Systems
Question 4 Explanation:
\begin{aligned} L^{-1}\left ( \frac{s+3}{s^2+2s+1} \right )&=L^{-1}\left ( \frac{s+1+2}{(s+1)^2} \right )\\ &=L^{-1}\left ( \frac{1}{s+1}+\frac{2}{(s+1)^2} \right )\\ &=e^{-t}+2te^{-t} \end{aligned}
 Question 5
Consider a linear time-invariant system with transfer function
$H(s)=\frac{1}{(s+1)}$
If the input is cos(t) and the steady state output is $A \cos (t+\alpha )$, then the value of A is _________.
 A 0.7 B 0.26 C 0.96 D 1.2
GATE EE 2016-SET-2   Signals and Systems
Question 5 Explanation:
\begin{aligned} H(s)&=\frac{1}{(s+1)}\\ \text{Put, }s&=j\omega \\ H(j\omega)&=\frac{1}{j\omega+1}\\ |H(j\omega)|&=\frac{1}{\sqrt{\omega ^2 +1}}\\ \because \; \text{ input }x(t)&=\cos (t)\\ \text{Here, } \omega &=1 \text{rad/sec}\\ \text{and }|x(t)|&=1\\ \text{hence, }&\text{steady state output}\\ y(t)&=|H(j\omega)|_{\omega=1} \cos (t+\angle H(j\omega)|_{\omega =1})\\ A&=|H(j\omega)|_{\omega =1}=\frac{1}{\sqrt{2}}=0.707 \end{aligned}
 Question 6
The transfer function of a system is $\frac{Y(s)}{R(s)}=\frac{s}{s+2}$. The steady state output y(t) is $A \cos (2t+\varphi)$ for the input cos(2t). The values of A and $\varphi$, respectively are
 A $\frac{1}{\sqrt{2}},-45^{\circ}$ B $\frac{1}{\sqrt{2}},+45^{\circ}$ C $\sqrt{2},-45^{\circ}$ D $\sqrt{2},+45^{\circ}$
GATE EE 2016-SET-1   Signals and Systems
Question 6 Explanation:
\begin{aligned} \frac{Y(s)}{R(s)}&=\frac{s}{s+2} \\ y(t)&=A \cos (2t+\phi ), \\ r(t)&=\cos 2t \\ \because \;H(s) &=\frac{s}{(s+2)} \\ H(j\omega )&=\frac{j\omega }{j\omega +2} \\ |H(j\omega )| &=\frac{\omega }{\sqrt{\omega ^2+4 }} \\ \angle H(j\omega ) &=90^{\circ} -\tan ^{-1}\left ( \frac{\omega }{2} \right ) \\ \because \; \omega &= 2 \text{ (as given)}\\ |H(j\omega )| &=\frac{2}{\sqrt{4+4}}=\frac{1}{\sqrt{2}} \\ |H(j\omega )| &=90^{\circ} -\tan ^{-1}(1)=45^{\circ} \\ \because \; \text{hence, }A &=1 \times |H(j\omega )|_{\omega =2}\\ &=1 \times \frac{1}{\sqrt{2}}=0.707\\ \phi &= 45^{\circ} \end{aligned}
 Question 7
The Laplace Transform of $f(t)=e^{2t}sin(5t)u(t)$ is
 A $\frac{5}{s^{2}-4s+29}$ B $\frac{5}{s^{2}+5}$ C $\frac{s-2}{s^{2}-4s+29}$ D $\frac{5}{s+5}$
GATE EE 2016-SET-1   Signals and Systems
Question 7 Explanation:
Laplace transform of $\sin 5t u(t)\rightarrow \frac{5}{s^2+25}$
$e^{2t}\sin 5t u(t)\rightarrow \frac{5}{(s-2)^2+25}=\frac{5}{s^2-4s+29}$
 Question 8
The Laplace transform of $f(t)=2\sqrt{t/\pi }$ is $s^{-3/2}$. The Laplace transform of $g(t)=\sqrt{1/\pi t}$ is
 A $3s^{-5/2}/2$ B $s^{-1/2}$ C $s^{1/2}$ D $s^{3/2}$
GATE EE 2015-SET-2   Signals and Systems
Question 8 Explanation:
Given that,
$f(t)=2\sqrt{\frac{t}{\pi}}\rightleftharpoons F(s)=s^{-3/2}$
By using property of differentiation In time,
\begin{aligned} \frac{df(t)}{dt}&\rightleftharpoons sF(s) \\ \Rightarrow \; \frac{2}{\sqrt{\pi}}\cdot \frac{1}{2}t^{-1/2}&\rightleftharpoons sF(s) \\ \Rightarrow \; \frac{1}{\sqrt{\pi t}}&\rightleftharpoons s\cdot s^{-3/2} \\ \Rightarrow \; \frac{1}{\sqrt{\pi t}}&\rightleftharpoons s^{-1/2} \end{aligned}
 Question 9
Consider an LTI system with transfer function
$H(s)=\frac{1}{s(s+4)}$
If the input to the system is cos(3t) and the steady state output is Asin(3t+$\alpha$) , then the value of A is
 A $1/30$ B $1/15$ C $3/4$ D $4/3$
GATE EE 2014-SET-2   Signals and Systems
Question 9 Explanation:
Given,
\begin{aligned} H(s)&=\frac{1}{s(s+4)}\\ r(t)&=\text{input}= \cos (3t)=\cos \omega t\\ \therefore \; \omega &=3 \text{ rad/s}\\ H(j\omega)&=\frac{1}{(j\omega)(j\omega+4)}\\ \text{Now, }|H(j\omega)|&=\frac{1}{\omega \sqrt{\omega^4+4^2}}\\ &=\frac{1}{3\sqrt{25}}=\frac{1}{15} \;\;\;(at\; \omega=3)\\ \angle H(j\omega)&=-90^{\circ}-\tan ^{-1}\frac{\omega}{4}\\ &=-90^{\circ}-\tan ^{-1}\frac{3}{4}=-126.86^{\circ}\\ \therefore \; c(t)&=\frac{1}{15} \cos (3t-126.86^{\circ})\\ &=\frac{1}{15} \sin (3t-36.86^{\circ})\;\;...(i)\\ c(t)&=A \sin (3t+\alpha )\;\;...(ii)\\ \text{Comparing }&\text{ eq. (i) and (ii), we have,}\\ A&=\frac{1}{15} \end{aligned}
 Question 10
Which one of the following statements is NOT TRUE for a continuous time causal and stable LTI system?
 A All the poles of the system must lie on the left side of the $j \omega$ axis B Zeros of the system can lie anywhere in the s-plane C All the poles must lie within |s| = 1 D All the roots of the characteristic equation must be located on the left side of the $j \omega$ axis.
GATE EE 2013   Signals and Systems
Question 10 Explanation:
All poles must lie within |Z|=1
There are 10 questions to complete. 