Table of Contents
What is Laplace Transform?
The Laplace transform is a mathematical tool that converts complex differential equations into simpler algebraic equations. It helps analyze and solve problems in various fields like electrical engineering, control systems, and signal processing.
In simple words, the Laplace transform takes a function of time, like f(t), and converts it into a function of a complex variable, like F(s). This allows us to work with the transformed function, which is often easier to manipulate and solve.
Definition: The Laplace transform converts a function of a real variable to a function of a complex variable.
Laplace Transform Example
Let’s say we have a function f(t) = eat, where a is a constant.
The Laplace transform of f(t), denoted as F(s), is given by:
F(s) = ∫0∞ e-st ⋅ eat dt
F(s) = ∫0∞ e(a-s)t dt
F(s) = [e(a-s)t / (a-s)]0∞
F(s) = limt → ∞ [e(a-s)t / (a-s)] – limt → 0 [e(a-s)t / (a-s)]
F(s) = 1 / (s-a)
So, the Laplace transform of f(t) = eat is F(s) = 1 / (s-a).
Laplace Transform MCQ Questions and Answers
Here are the 20 Laplace transform questions and answers from basics to advanced level:
1. What is the Laplace transform of the unit step function \( u(t) \)?
- A) \( \frac{1}{s} \)
- B) \( \frac{1}{s^2} \)
- C) \( \frac{1}{s^3} \)
- D) \( \frac{1}{s^4} \)
Answer: A) \( \frac{1}{s} \)
Solution: The Laplace transform of \( u(t) \) is \( \frac{1}{s} \) since \( \mathcal{L}\{u(t)\} = \int_{0}^{\infty} e^{-st} dt = \frac{1}{s} \) for \( \text{Re}(s) > 0 \).
2. Which property of the Laplace transform states that \( \mathcal{L}\{e^{at}f(t)\} = F(s-a) \)?
- A) Linearity
- B) Time Shifting
- C) Differentiation
- D) Integration
Answer: B) Time Shifting
Solution: According to the time shifting property of Laplace transform, \( \mathcal{L}\{e^{at}f(t)\} = F(s-a) \).
3. What is the Laplace transform of \( t \)?
- A) \( \frac{1}{s^2} \) – \( \frac{1}{s} \)
- B) \( \frac{1}{s^3} \) – \( \frac{1}{s^2} \)
- C) \( \frac{1}{s^4} \) – \( \frac{1}{s^3} \)
- D) \( \frac{1}{s^2} \) – \( \frac{1}{s^3} \)
Answer: A) \( \frac{1}{s^2} \) – \( \frac{1}{s} \)
Solution: \( \mathcal{L}\{t\} = \int_0^\infty t \cdot e^{-st} dt = \frac{1}{s^2} – \frac{1}{s} \) using integration by parts.
4. Which property of the Laplace transform states that \( \mathcal{L}\{f'(t)\} = sF(s) – f(0) \)?
- A) Linearity
- B) Time Shifting
- C) Differentiation
- D) Integration
Answer: C) Differentiation
Solution: According to the differentiation property of Laplace transform, \( \mathcal{L}\{f'(t)\} = sF(s) – f(0) \).
5. What is the Laplace transform of \( e^{at} \)?
- A) \( \frac{1}{s-a} \)
- B) \( \frac{1}{s+a} \)
- C) \( \frac{1}{s^2-a^2} \)
- D) \( \frac{1}{s^2+a^2} \)
Answer: A) \( \frac{1}{s-a} \)
Solution: \( \mathcal{L}\{e^{at}\} = \int_0^\infty e^{-st} \cdot e^{at} dt = \frac{1}{s-a} \).
6. What is the Laplace transform of the Dirac delta function \( \delta(t) \)?
- A) \( 1 \)
- B) \( e \)
- C) \( 0 \)
- D) \( \infty \)
Answer: A) \( 1 \)
Solution: The Laplace transform of the Dirac delta function \( \delta(t) \) is \( 1 \) since \( \mathcal{L}\{\delta(t)\} = \int_{0}^{\infty} \delta(t) \cdot e^{-st} dt = 1 \).
7. Which property of the Laplace transform states that \( \mathcal{L}\{f(at)\} = \frac{1}{a}F\left(\frac{s}{a}\right) \)?
- A) Linearity
- B) Time Scaling
- C) Differentiation
- D) Time Reversal
Answer: B) Time Scaling
Solution: According to the time scaling property of Laplace transform, \( \mathcal{L}\{f(at)\} = \frac{1}{a}F\left(\frac{s}{a}\right) \).
8. What is the Laplace transform of \( t^2 \)?
- A) \( \frac{2}{s^3} \)
- B) \( \frac{2}{s^2} \)
- C) \( \frac{2}{s} \)
- D) \( \frac{2}{s^4} \)
Answer: B) \( \frac{2}{s^2} \)
Solution: \( \mathcal{L}\{t^2\} = \int_{0}^{\infty} t^2 \cdot e^{-st} dt = \frac{2}{s^3} \).
9. Which property of the Laplace transform states that \( \mathcal{L}\{tf(t)\} = -F'(s) \)?
- A) Linearity
- B) Differentiation
- C) Integration
- D) Time Scaling
Answer: B) Differentiation
Solution: According to the differentiation property of Laplace transform, \( \mathcal{L}\{tf(t)\} = -F'(s) \).
10. What is the Laplace transform of \( e^{-at} \sin(bt) \)?
- A) \( \frac{b}{(s+a)^2 + b^2} \)
- B) \( \frac{a}{(s+a)^2 + b^2} \)
- C) \( \frac{s-a}{(s-a)^2 + b^2} \)
- D) \( \frac{s-b}{(s-a)^2 + b^2} \)
Answer: A) \( \frac{b}{(s+a)^2 + b^2} \)
Solution: \( \mathcal{L}\{e^{-at} \sin(bt)\} = \frac{b}{(s+a)^2 + b^2} \).
11. Which property of the Laplace transform states that \( \mathcal{L}\{f(at)\} = \frac{1}{a}F\left(\frac{s}{a}\right) \)?
- A) Linearity
- B) Time Scaling
- C) Differentiation
- D) Time Reversal
Answer: B) Time Scaling
Solution: According to the time scaling property of Laplace transform, \( \mathcal{L}\{f(at)\} = \frac{1}{a}F\left(\frac{s}{a}\right) \).
12. What is the Laplace transform of the function \( f(t) = e^{-at}\sin(bt) \)?
- A) \( \frac{b}{(s+a)^2 + b^2} \)
- B) \( \frac{a}{(s+a)^2 + b^2} \)
- C) \( \frac{s-a}{(s-a)^2 + b^2} \)
- D) \( \frac{s-b}{(s-a)^2 + b^2} \)
Answer: A) \( \frac{b}{(s+a)^2 + b^2} \)
Solution: \( \mathcal{L}\{e^{-at}\sin(bt)\} = \frac{b}{(s+a)^2 + b^2} \).
13. Which property of the Laplace transform states that \( \mathcal{L}\{tf(t)\} = -F'(s) \)?
- A) Linearity
- B) Differentiation
- C) Integration
- D) Time Scaling
Answer: B) Differentiation
Solution: According to the differentiation property of Laplace transform, \( \mathcal{L}\{tf(t)\} = -F'(s) \).
14. What is the Laplace transform of the function \( f(t) = t^2 \)?
- A) \( \frac{2}{s^3} \)
- B) \( \frac{2}{s^2} \)
- C) \( \frac{2}{s} \)
- D) \( \frac{2}{s^4} \)
Answer: B) \( \frac{2}{s^2} \)
Solution: \( \mathcal{L}\{t^2\} = \int_{0}^{\infty} t^2 \cdot e^{-st} dt = \frac{2}{s^3} \).
15. Which property of the Laplace transform states that \( \mathcal{L}\{\int_{0}^{t} f(\tau) d\tau\} = \frac{F(s)}{s} \)?
- A) Linearity
- B) Differentiation
- C) Integration
- D) Time Scaling
Answer: C) Integration
Solution: According to the integration property of Laplace transform, \( \mathcal{L}\{\int_{0}^{t} f(\tau) d\tau\} = \frac{F(s)}{s} \).
16. What is the Laplace transform of the function \( f(t) = \sinh(at) \)?
- A) \( \frac{a}{s^2 + a^2} \)
- B) \( \frac{s}{s^2 + a^2} \)
- C) \( \frac{1}{s^2 – a^2} \)
- D) \( \frac{a}{s^2 – a^2} \)
Answer: A) \( \frac{a}{s^2 + a^2} \)
Solution: \( \mathcal{L}\{\sinh(at)\} = \frac{a}{s^2 + a^2} \).
17. Which property of the Laplace transform states that \( \mathcal{L}\{f'(t)\} = sF(s) – f(0) \)?
- A) Linearity
- B) Time Shifting
- C) Differentiation
- D) Integration
Answer: C) Differentiation
Solution: According to the differentiation property of Laplace transform, \( \mathcal{L}\{f'(t)\} = sF(s) – f(0) \).
18. What is the Laplace transform of the function \( f(t) = \cosh(at) \)?
- A) \( \frac{s}{s^2 – a^2} \)
- B) \( \frac{a}{s^2 + a^2} \)
- C) \( \frac{s}{s^2 + a^2} \)
- D) \( \frac{a}{s^2 – a^2} \)
Answer: C) \( \frac{s}{s^2 + a^2} \)
Solution: \( \mathcal{L}\{\cosh(at)\} = \frac{s}{s^2 + a^2} \).
19. Which property of the Laplace transform states that \( \mathcal{L}\{\int_{0}^{t} f(\tau) d\tau\} = \frac{F(s)}{s} \)?
- A) Linearity
- B) Differentiation
- C) Integration
- D) Time Scaling
Answer: C) Integration
Solution: According to the integration property of Laplace transform, \( \mathcal{L}\{\int_{0}^{t} f(\tau) d\tau\} = \frac{F(s)}{s} \).
20. What is the Laplace transform of the function \( f(t) = e^{at}\cos(bt) \)?
- A) \( \frac{s+a}{(s+a)^2 + b^2} \)
- B) \( \frac{s-a}{(s-a)^2 + b^2} \)
- C) \( \frac{s-b}{(s-a)^2 + b^2} \)
- D) \( \frac{s}{(s-a)^2 + b^2} \)
Answer: A) \( \frac{s+a}{(s+a)^2 + b^2} \)
Solution: \( \mathcal{L}\{e^{at}\cos(bt)\} = \frac{s+a}{(s+a)^2 + b^2} \).
FAQs on Laplace Transform MCQ
Who discovered Laplace transform?
The Laplace transform was first used by Pierre-Simon Laplace, a French mathematician and astronomer, in his work on probability theory. He developed the transform as a tool for solving differential equations and analyzing complex systems.
What is the Laplace of 1?
The Laplace transform of 1 is 1/s. This is because the Laplace transform of a function is defined as the integral of the function multiplied by e^(-st) from 0 to infinity, and the integral of 1 is simply 1/s.
What is Laplace transform in simple language?
The Laplace transform is a mathematical tool that helps us solve complex problems by converting them into simpler algebraic equations. It is used to analyze systems and signals in the frequency domain, making it easier to understand and predict their behavior.
What are the applications of Laplace transform?
The Laplace transform has many applications in various fields, including control systems, electrical engineering, signal processing, and probability theory. It is used to analyze and design systems, solve differential equations, and model complex phenomena.
What is the principle of Laplace transform?
The principle of the Laplace transform is to convert a function of time into a function of a complex variable, usually interpreted as complex frequency. This allows us to analyze and solve problems in the frequency domain, making it easier to understand and predict the behavior of complex systems.
