Question 1:
\[
\text{What is the Laplace transform of } \delta(t) \text{, where } \delta(t) \text{ is the Dirac delta function?}
\]
\[
\text{(a) } 1, \quad \text{(b) } 0, \quad \text{(c) } \infty, \quad \text{(d) } \text{undefined}
\]
Answer: A
Question 2:
\[
\text{What is the Laplace transform of } e^{at}?
\]
\[
\text{(a) } \frac{1}{s-a}, \quad \text{(b) } \frac{1}{s+a}, \quad \text{(c) } \frac{s}{s^2 + a^2}, \quad \text{(d) } \frac{a}{s^2 + a^2}
\]
Answer: A
Question 3:
\[
\text{What is the Laplace transform of } \sin(at)?
\]
\[
\text{(a) } \frac{a}{s^2 + a^2}, \quad \text{(b) } \frac{s}{s^2 + a^2}, \quad \text{(c) } \frac{1}{s+a}, \quad \text{(d) } \frac{1}{s^2 + a^2}
\]
Answer: A
Question 4:
\[
\text{What is the Laplace transform of } \cos(at)?
\]
\[
\text{(a) } \frac{s}{s^2 + a^2}, \quad \text{(b) } \frac{a}{s^2 + a^2}, \quad \text{(c) } \frac{1}{s^2 + a^2}, \quad \text{(d) } \frac{s}{s+a}
\]
Answer: A
Question 5:
\[
\text{What is the Laplace transform of } t^n?
\]
\[
\text{(a) } \frac{n!}{s^{n+1}}, \quad \text{(b) } \frac{1}{s^{n+1}}, \quad \text{(c) } \frac{n}{s^{n+1}}, \quad \text{(d) } \frac{1}{n!s^{n+1}}
\]
Answer: A
Question 6:
\[
\text{What is the Laplace transform of } \frac{1}{t}?
\]
\[
\text{(a) } \ln(s), \quad \text{(b) } \frac{1}{s}, \quad \text{(c) } \frac{1}{s^2}, \quad \text{(d) } \text{undefined}
\]
Answer: D
Question 7:
\[
\text{What is the inverse Laplace transform of } \frac{1}{s(s+1)}?
\]
\[
\text{(a) } 1 – e^{-t}, \quad \text{(b) } e^{-t}, \quad \text{(c) } 1 + e^{-t}, \quad \text{(d) } 1 – e^{t}
\]
Answer: A
Question 8:
\[
\text{The Laplace transform of the derivative } \frac{d}{dt}[f(t)] \text{ is:}
\]
\[
\text{(a) } sF(s) – f(0), \quad \text{(b) } sF(s) + f(0), \quad \text{(c) } F(s) – f(0), \quad \text{(d) } sF(s)
\]
Answer: A
Question 9:
\[
\text{The Laplace transform of a step function } u(t-a) \text{ is:}
\]
\[
\text{(a) } \frac{1}{s}, \quad \text{(b) } \frac{e^{-as}}{s}, \quad \text{(c) } \frac{1}{s^2}, \quad \text{(d) } \frac{1}{s+a}
\]
Answer: B
Question 10:
\[
\text{The Laplace transform of } \int_0^t f(\tau) d\tau \text{ is:}
\]
\[
\text{(a) } \frac{F(s)}{s}, \quad \text{(b) } sF(s), \quad \text{(c) } F(s), \quad \text{(d) } F(s) + \frac{1}{s}
\]
Answer: A
