## Laplace Transform

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

1998-1-1-math

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

2004-1-math

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

2014-2-math

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

2016-2-math

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

EC-2012-11-math

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

• $$\dfrac {-s}{(s^2+s+1)^2}$$

• $$\dfrac {-(2s+1)}{(s^2+s+1)^2}$$

• $$\dfrac {s}{(s^2+s+1)^2}$$

• $$\dfrac {2s+1}{(s^2+s+1)^2}$$

[Index]

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

2007-27-math

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

2008-22-math

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

2017-27-math

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

1994-4-a-math

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

 (I) $$\cosh (at)$$ (A) $$a/(s^2+a^2)$$ (II) $$\sinh (at)$$ (B) $$a/(s^2-a^2)$$ (C) $$s/(s^2-a^2)$$ (D) $$s/(s^2+a^2)$$

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

2001-4-math

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: $\mathcal{L}\left[\sin(\omega t)\right] = \frac{\omega}{s^2+\omega^2}$

[Index]