MCQ on Laplace Transform

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Welcome to the “MCQ on Laplace Transform” quiz. The Laplace Transform is a powerful math tool. It turns complex differential equations into simpler algebraic ones. Engineers and scientists use it a lot. It helps in control systems, signal processing, and circuit analysis. This quiz will test your knowledge.

You will see questions about Laplace Transform properties. You will answer questions about common functions and their transformations.

Get ready to learn and test your skills!

1. What is the Laplace Transform of \( f(t) = e^{2t} \) ?

A. \( \frac{1}{s+2} \) B. \( \frac{1}{s-2} \) C. \( \frac{1}{s+2} \) D. \( \frac{1}{s-3} \)

2. The Laplace Transform of \( t^n \) is?

A. \( \frac{n!}{s^{n+1}} \) B. \( \frac{1}{s^{n+1}} \) C. \( \frac{1}{(n+1)!} \) D. \( \frac{(n+1)!}{s^n} \)

3. What is the inverse Laplace Transform of \( \frac{1}{s-3} \)?

A. \( e^{3s} \) B. \( e^{3t} \) C. \( e^{-3t} \) D. \( e^{2t} \)

4. The Laplace Transform of \( \cos(at) \) is?

A. \( \frac{a}{s^2 + a^2} \) B. \( \frac{s}{s^2 + a^2} \) C. \( \frac{s}{s^2 + a^2} \) D. \( \frac{a}{s^2 – a^2} \)

5. What is the Laplace Transform of \( \delta(t) \) where \( \delta(t) \) is the Dirac delta function?

A. \( 1 \) B. \( 0 \) C. \( s \) D. \( 1 \)

6. Which property of the Laplace Transform is used to solve differential equations?

A. Linearity B. Differentiation C. Convolution D. Shifting

7. What is the Laplace Transform of \( \sin(bt) \) ?

A. \( \frac{b}{s+b} \) B. \( \frac{b}{s^2 + b^2} \) C. \( \frac{s}{s^2 – b^2} \) D. \( \frac{s}{s^2 + b^2} \)

8. The Laplace Transform of \( f(t) \ast g(t) \) is?

A. \( F(s) + G(s) \) B. \( F(s)G(s) \) C. \( F(s) – G(s) \) D. \( F(s)/G(s) \)

9. What is the initial value theorem of Laplace Transform?

A. \( \lim_{{t \to 0^+}} f(t) = \lim_{{s \to \infty}} sF(s) \) B. \( \lim_{{t \to \infty}} f(t) = \lim_{{s \to \infty}} sF(s) \) C. \( \lim_{{t \to 0^+}} f(t) = \lim_{{s \to 0}} sF(s) \) D. \( \lim_{{t \to \infty}} f(t) = \lim_{{s \to 0}} sF(s) \)

10. The Laplace Transform of \( \cosh(at) \) is?

A. \( \frac{s^2}{s^2 – a^2} \) B. \( \frac{s}{s^2 – a^2} \) C. \( \frac{a}{s^2 – a^2} \) D. \( \frac{s}{s^2 + a^2} \)

11. What is the Laplace Transform of the unit step function \( u(t) \)?

A. \( \frac{1}{s} \) B. \( \frac{s}{1} \) C. \( s \) D. \( \frac{1}{s^2} \)

12. The final value theorem of Laplace Transform states?

A. \( \lim_{{t \to \infty}} f(t) = \lim_{{s \to \infty}} sF(s) \) B. \( \lim_{{t \to 0^+}} f(t) = \lim_{{s \to 0}} sF(s) \) C. \( \lim_{{t \to \infty}} f(t) = \lim_{{s \to 0}} sF(s) \) D. \( \lim_{{t \to 0^+}} f(t) = \lim_{{s \to \infty}} sF(s) \)

13. What is the Laplace Transform of \( e^{at} \cos(bt) \) ?

A. \( \frac{s-a}{(s-a)^2 – b^2} \) B. \( \frac{s+a}{(s-a)^2 + b^2} \) C. \( \frac{s-a}{(s-a)^2 + b^2} \) D. \( \frac{s+a}{(s+a)^2 – b^2} \)

14. Which of the following is the region of convergence (ROC) for the Laplace Transform of a causal signal?

A. \( \Re(s) > \sigma \) B. \( \Re(s) < \sigma \) C. \( \Re(s) = \sigma \) D. \( \Im(s) > \sigma \)

15. The Laplace Transform of a constant \( C \) is?

A. \( C \) B. \( \frac{C}{s} \) C. \( Cs \) D. \( \frac{C}{s^2} \)

16. What does the Laplace Transform of \( e^{-at} f(t) \) result in?

A. \( F(s+a) \) B. \( F(s-a) \) C. \( F(s) \) D. \( F(-s+a) \)

17. The Laplace Transform is particularly useful for analyzing which type of systems?

A. Non-linear systems B. Static systems C. Linear time-invariant systems D. Dynamic systems

18. What is the Laplace Transform of \( \sinh(at) \)?

A. \( \frac{a}{s^2 – a^2} \) B. \( \frac{s}{s^2 + a^2} \) C. \( \frac{s}{s^2 – a^2} \) D. \( \frac{a}{s^2 + a^2} \)

19. The Laplace Transform of \( \int_0^t f(\tau) \, d\tau \) is?

A. \( F(s) \) B. \( \frac{F(s)}{s} \) C. \( sF(s) \) D. \( \frac{1}{F(s)} \)

20. Which of the following is a common application of the Laplace Transform in engineering?

A. Circuit analysis B. Fluid dynamics C. Aerodynamics D. Control systems

21. What is the Laplace Transform of \( t \cdot e^{at} \)?

A. \( \frac{1}{(s-a)^2} \) B. \( \frac{1}{(s-a)^2} \) C. \( \frac{1}{s^2} \) D. \( \frac{1}{(s+a)^2} \)

22. What is the inverse Laplace Transform of \( \frac{s}{s^2 + \omega^2} \)?

A. \( \cosh(\omega t) \) B. \( \cos(\omega t) \) C. \( \sin(\omega t) \) D. \( \sinh(\omega t) \)

23. The Laplace Transform of \( e^{at} \sin(bt) \) is?

A. \( \frac{s+b}{(s-a)^2 + b^2} \) B. \( \frac{s-a}{(s+a)^2 – b^2} \) C. \( \frac{b}{(s-a)^2 + b^2} \) D. \( \frac{a}{(s-a)^2 + b^2} \)

24. What is the Laplace Transform of \( f(t) \cdot g(t) \)?

A. \( F(s)G(s) \) B. \( \int_0^s F(\sigma)G(s-\sigma) \, d\sigma \) C. None of the above D. \( F(s) + G(s) \)

25. The Laplace Transform is a linear operator. This means that for any functions \( f(t) \) and \( g(t) \), and any constants \( a \) and \( b \):

A. \( \mathcal{L}\{af(t) + bg(t)\} = a\mathcal{L}\{f(t)\} \cdot b\mathcal{L}\{g(t)\} \) B. \( \mathcal{L}\{af(t) + bg(t)\} = a\mathcal{L}\{f(t)\} + b\mathcal{L}\{g(t)\} \) C. \( \mathcal{L}\{af(t) + bg(t)\} = a\mathcal{L}\{f(t) + g(t)\} \) D. \( \mathcal{L}\{af(t) + bg(t)\} = \mathcal{L}\{a\cdot f(t) + b\cdot g(t)\} \)

26. Which of the following expressions is the Laplace Transform of \( \delta(t – a) \), where \( \delta \) is the Dirac delta function?

A. \( e^{-as} \) B. \( e^{-as} \) C. \( e^{as} \) D. \( 1 \)

27. The Laplace Transform of the Heaviside step function \( u(t-a) \) is?

A. \( \frac{1}{s} \cdot e^{-as} \) B. \( \frac{e^{-as}}{s} \) C. \( \frac{e^{-as}}{s} \) D. \( \frac{e^{as}}{s} \)

28. Which of the following is a bilateral Laplace Transform?

A. \( \int_0^\infty e^{-st} f(t) \, dt \) B. \( \int_0^\infty e^{-st} f(t) \, dt \) C. \( \int_{-\infty}^\infty e^{-st} f(t) \, dt \) D. \( \int_{-\infty}^0 e^{-st} f(t) \, dt \)

29. The Laplace Transform can be used to solve which type of differential equations?

A. Ordinary differential equations B. Partial differential equations C. Both ordinary and partial differential equations D. Difference equations

30. Which function’s Laplace Transform is given by \( \frac{s}{s^2 + a^2} \)?

A. \( \cosh(at) \) B. \( \sinh(at) \) C. \( \sin(at) \) D. \( \cos(at) \)

Review Your Answers Above.

Conclusion

Thanks for taking the “MCQ on Laplace Transform” quiz. We hope you enjoyed it. These questions were made to help you understand the Laplace Transform better.

Keep exploring and practicing.

We wish you success in your studies and career.

Abdur Rohman

Abdur Rohman is an Electrical Engineer from Charaideo, Assam, who wears multiple hats as a part-time teacher, blogger, entrepreneur, and digital marketer. Passionate about education, he founded The Assam School blog to provide free, comprehensive textbook solutions, MCQs (Multiple Choice Questions), and other academic content for students from Class V to XII.

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MCQ Quiz on Laplace Transform | Test Your Understanding