GATE-CH-1998-1-1-math-1mark

The Laplace transform of the function $e^{-at}$ has the form:

• ${\displaystyle \frac{1}{s+1}}$

• ${\displaystyle \frac{1}{s(s+a)}}$

• ${\displaystyle \frac{a}{s}}$

• ${\displaystyle \frac{1}{s+a}}$

GATE-CH-2004-1-math-1mark

The inverse Laplace transform of the function ${\displaystyle f(s)=\frac{1}{s(1+s)}}$ is

• $1+e^t$

• $1-e^t$

• $1+e^{-t}$

• $1-e^{-t}$

GATE-CH-2014-2-math-1mark

For the time domain function $f(t)=t^2$, which ONE of the following is the Laplace transform of ${\displaystyle {\int }_0^{t}f(t)dt}$?

• ${\displaystyle \frac{3}{s^4}}$

• ${\displaystyle \frac{1}{4s^2}}$

• ${\displaystyle \frac{2}{s^3}}$

• ${\displaystyle \frac{2}{s^4}}$

GATE-CH-2016-2-math-1mark

The Laplace transform of $e^{at}\sin (bt)$ is

• ${\displaystyle \frac{b}{(s-a{)}^2+b^2}}$

• ${\displaystyle \frac{(s-a)}{(s-a{)}^2+b^2}}$

• ${\displaystyle \frac{(s-a)}{(s-a{)}^2-b^2}}$

• ${\displaystyle \frac{b}{(s-a{)}^2-b^2}}$

GATE-CH-EC-2012-11-math-1mark

The unilateral Laplace transform of $f(t)$ is ${\displaystyle \frac{1}{s^2+s+1}}$. The unilateral Laplace transform of $tf(t)$ is

• ${\displaystyle \frac{-s}{(s^2+s+1{)}^2}}$

• ${\displaystyle \frac{-(2s+1)}{(s^2+s+1{)}^2}}$

• ${\displaystyle \frac{s}{(s^2+s+1{)}^2}}$

• ${\displaystyle \frac{2s+1}{(s^2+s+1{)}^2}}$

GATE-CH-2007-27-math-2mark

The Laplace transform of ${\displaystyle f(t)=\frac{1}{\sqrt{t}}}$ is

• ${\displaystyle \sqrt{\frac{\pi }{s}}}$

• ${\displaystyle \frac{1}{\sqrt{s}}}$

• ${\displaystyle \frac{1}{s^{3/2}}}$

• does not exist

GATE-CH-2008-22-math-2mark

The Laplace transform of the function $f(t)=t\sin t$ is

• ${\displaystyle \frac{2s}{(s^2+1{)}^2}}$

• ${\displaystyle \frac{1}{s^2(s^2+1)}}$

• ${\displaystyle \frac{1}{s^2}+\frac{1}{(s^2+1)}}$

• ${\displaystyle \frac{1}{(s-1{)}^2+1}}$

GATE-CH-2017-27-math-2mark

The Laplace transform of function is ${\displaystyle \frac{s+1}{s(s+2)}}$. The initial and final values, respectively, of the function are

• 0 and 1

• 1 and ${\displaystyle \frac{1}{2}}$

• ${\displaystyle \frac{1}{2}}$ and 1

• ${\displaystyle \frac{1}{2}}$ and 0

GATE-CH-1994-4-a-math-1mark

Match the items in the left column with the appropriate items in the right column.

• Answer

GATE-CH-2001-4-math-5mark

Laplace transforms:

1. Show that the Laplace transform of $e^{\omega t}$ is $$\mathcal{L}\left[e^{\omega t}\right]=\frac{1}{s-\omega }$$

2. Show from (a) that: $$\mathcal{L}[\sinh (\omega t)]=\frac{\omega }{s^2-{\omega }^2}$$

3. Show from (b) that: $$\mathcal{L}[\sin (\omega t)]=\frac{\omega }{s^2+{\omega }^2}$$

• Answer