Laplace transform mcq Set-1

Attempt Laplace transform mcq Set-1 and checkout complete solution below

Here’s complete solution of 20 MCQs on the Laplace Transform:

Question 1

What is the Laplace Transform of ( f(t) = t^n ) where ( n ) is a positive integer?

Solution:

The Laplace Transform of ( f(t) = t^n ) is given by:
[ \mathcal{L}{t^n} = \int_0^\infty t^n e^{-st} \, dt ]

Using the formula for the Laplace Transform of ( t^n ):
[ \mathcal{L}{t^n} = \frac{n!}{s^{n+1}} ]

So, the answer is:
A) ( \frac{n!}{s^{n+1}} )

Question 2

What is the Laplace Transform of ( f(t) = \cos(at) )?

Solution:

The Laplace Transform of ( \cos(at) ) is:
[ \mathcal{L}{\cos(at)} = \int_0^\infty \cos(at) e^{-st} \, dt ]

Using the standard formula for the Laplace Transform:
[ \mathcal{L}{\cos(at)} = \frac{s}{s^2 + a^2} ]

So, the answer is:
B) ( \frac{s}{s^2 + a^2} )

Question 3

The Laplace Transform of ( e^{at} \cdot \sin(bt) ) is:

Solution:

The Laplace Transform of ( e^{at} \cdot \sin(bt) ) can be computed using the shifting theorem:
[ \mathcal{L}{e^{at} \sin(bt)} = \frac{b}{(s – a)^2 + b^2} ]

So, the answer is:
A) ( \frac{b}{(s – a)^2 + b^2} )

Question 4

What is the Laplace Transform of ( f(t) = t \cdot e^{bt} )?

Solution:

To find the Laplace Transform of ( t \cdot e^{bt} ), use the derivative property of the Laplace Transform:
[ \mathcal{L}{t \cdot e^{bt}} = -\frac{d}{ds} \left( \frac{1}{s – b} \right) = \frac{1}{(s – b)^2} ]

So, the answer is:
B) ( \frac{1}{(s – b)^2} – \frac{1}{s – b} )

Question 5

Which of the following is the Laplace Transform of

( f(t) = \delta(t – a) ) where ( \delta ) is the Dirac delta function?

Solution:

The Laplace Transform of the Dirac delta function ( \delta(t – a) ) is:
[ \mathcal{L}{\delta(t – a)} = e^{-as} ]

So, the answer is:
A) ( e^{-as} )

Question 6

What is the Laplace Transform of ( f(t) = \sin(at) \cdot \cos(bt) )?

Solution:

Using the trigonometric identity:
[ \sin(at) \cdot \cos(bt) = \frac{1}{2} [\sin((a+b)t) + \sin((a-b)t)] ]

The Laplace Transform of ( \sin((a+b)t) ) is ( \frac{a+b}{s^2 + (a+b)^2} ) and of ( \sin((a-b)t) ) is ( \frac{a-b}{s^2 + (a-b)^2} ). Combining them:

[ \mathcal{L}{\sin(at) \cdot \cos(bt)} = \frac{1}{2} \left( \frac{a+b}{s^2 + (a+b)^2} + \frac{a-b}{s^2 + (a-b)^2} \right) ]

So, the answer is:
C) ( \frac{(s^2 – a^2 – b^2)}{(s^2 + a^2)(s^2 + b^2)} )

Question 7

If ( \mathcal{L}{f(t)} = F(s) ), what is the Laplace Transform of ( f(t – a) \cdot u(t – a) ), where ( u(t – a) ) is the unit step function?

Solution:

Using the shifting theorem:
[ \mathcal{L}{f(t – a) \cdot u(t – a)} = e^{-as} F(s) ]

So, the answer is:
A) ( e^{-as} F(s) )

Question 8

What is the Laplace Transform of ( f(t) = t \cdot \cos(at) )?

Solution:

Using the formula for the Laplace Transform of ( t \cdot \cos(at) ):
[ \mathcal{L}{t \cdot \cos(at)} = \frac{s(s^2 – a^2) + 2a}{(s^2 + a^2)^2} ]

So, the answer is:
A) ( \frac{s(s^2 – a^2) + 2a}{(s^2 + a^2)^2} )

Question 9

What is the Laplace Transform of ( f(t) = \frac{1}{t} )?

Solution:

The Laplace Transform of ( \frac{1}{t} ) does not exist in the standard form as the Laplace Transform is usually defined for functions that grow at most polynomially. However, using advanced transforms, it can be related to the Gamma function:
[ \mathcal{L}\left{\frac{1}{t}\right} = -\log(s) ]

So, the answer is:
C) ( -\log(s) )

Question 10

The Laplace Transform of ( f(t) = e^{bt} \cdot \cos(at) ) is:

Solution:

Using the shifting theorem and formula for the Laplace Transform:
[ \mathcal{L}{e^{bt} \cos(at)} = \frac{s – b}{(s – b)^2 + a^2} ]

So, the answer is:
A) ( \frac{s – b}{(s – b)^2 + a^2} )

Question 11

What is the Laplace Transform of ( f(t) = \sinh(at) )?

Solution:

The Laplace Transform of ( \sinh(at) ) is:
[ \mathcal{L}{\sinh(at)} = \frac{a}{s^2 – a^2} ]

So, the answer is:
A) ( \frac{a}{s^2 – a^2} )

Question 12

The Laplace Transform of ( f(t) = \cosh(at) ) is:

Solution:

The Laplace Transform of ( \cosh(at) ) is:
[ \mathcal{L}{\cosh(at)} = \frac{s}{s^2 – a^2} ]

So, the answer is:
A) ( \frac{s}{s^2 – a^2} )

Question 13

If ( f(t) ) has the Laplace Transform ( F(s) ), what is the Laplace Transform of ( e^{at} \cdot f(t) )?

Solution:

Using the shifting theorem:
[ \mathcal{L}{e^{at} f(t)} = F(s – a) ]

So, the answer is:
A) ( F(s – a) )

Question 14

What is the Laplace Transform of ( f(t) = \cos(at) ) where ( a ) is a constant?

Solution:

As in Question 2, the Laplace Transform is:
[ \mathcal{L}{\cos(at)} = \frac{s}{s^2 + a^2} ]

So, the answer is:
B) ( \frac{s}{s^2 + a^2} )

Question 15

Which of the following is the Laplace Transform of ( t^n \cdot e^{bt} )?

Solution:

Using the formula for ( t^n \cdot e^{bt} ):
[ \mathcal{L}{t^n \cdot e^{bt}} = \frac{n!}{(s – b)^{n+1}} ]

So, the answer is:
A) ( \frac{n!}{(s – b)^{n+1}} )

Question 16

What is the Laplace Transform of ( f(t) = e^{-at} \cdot \sin(bt) )?

Solution:

Using the shifting theorem and formula:
[ \mathcal{L}{e^{-at} \sin(bt)\

} = \frac{b}{(s + a)^2 + b^2} ]

So, the answer is:
A) ( \frac{b}{(s + a)^2 + b^2} )

Question 17

The Laplace Transform of ( f(t) = \frac{1}{t^n} ) is:

Solution:

For ( t^{-n} ), the Laplace Transform is:
[ \mathcal{L}\left{t^{-n}\right} = \frac{\Gamma(1 – n)}{s^{1-n}} ]

Using the Gamma function (\Gamma), the correct formula is:
[ \frac{\Gamma(n – 1)}{s^n} ]

So, the answer is:
C) ( \frac{\Gamma(n)}{s^n} )

Question 18

If ( \mathcal{L}{f(t)} = F(s) ), what is the Laplace Transform of ( \frac{d}{dt} f(t) )?

Solution:

Using the differentiation property:
[ \mathcal{L}\left{\frac{d}{dt} f(t)\right} = s \cdot F(s) – f(0) ]

So, the answer is:
A) ( s \cdot F(s) – f(0) )

Question 19

What is the Laplace Transform of ( f(t) = \cos(at) \cdot e^{bt} )?

Solution:

Using the shifting theorem and formula:
[ \mathcal{L}{\cos(at) \cdot e^{bt}} = \frac{s – b}{(s – b)^2 + a^2} ]

So, the answer is:
A) ( \frac{s – b}{(s – b)^2 + a^2} )

Question 20

What is the Laplace Transform of ( f(t) = \sin(at) \cdot \cos(bt) )?

Solution:

Using the trigonometric identity and Laplace Transforms:
[ \sin(at) \cos(bt) = \frac{1}{2} [\sin((a+b)t) + \sin((a-b)t)] ]

The Laplace Transform of these terms combined gives:
[ \frac{(s^2 – a^2 – b^2)}{(s^2 + a^2)(s^2 + b^2)} ]

So, the answer is:
A) ( \frac{(s^2 – a^2 – b^2)}{(s^2 + a^2)(s^2 + b^2)} )

Feel free to use these detailed solutions to understand the process and formulas involved in calculating Laplace Transforms!