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

The Laplace transform of the function \(e^{-at}\) has the form:

• \(\dfrac {1}{s+1}\)

• \(\dfrac {1}{s(s+a)}\)

• \(\dfrac {a}{s}\)

• \(\dfrac {1}{s+a}\)

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

• \(\dfrac {b}{(s-a)^2+b^2}\)

• \(\dfrac {(s-a)}{(s-a)^2+b^2}\)

• \(\dfrac {(s-a)}{(s-a)^2-b^2}\)

• \(\dfrac {b}{(s-a)^2-b^2}\)

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

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

• \(\dfrac {-s}{(s^2+s+1)^2}\)

• \(\dfrac {-(2s+1)}{(s^2+s+1)^2}\)

• \(\dfrac {s}{(s^2+s+1)^2}\)

• \(\dfrac {2s+1}{(s^2+s+1)^2}\)

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 \(\dfrac {s+1}{s(s+2)}\). The initial and final values, respectively, of the function are

• 0 and 1

• 1 and \(\dfrac {1}{2}\)

• \(\dfrac {1}{2}\) and 1

• \(\dfrac {1}{2}\) 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: \[\mathcal{L}\left[\sinh(\omega t)\right] = \frac{\omega}{s^2-\omega^2}\]

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

• Answer