Which of the following statements about Laplace transforms and convolution is correct?

• A) Convolution in time domain corresponds to convolution in the Laplace domain.
• B) Multiplication in time domain corresponds to convolution in Laplace domain.
• C) Convolution and Laplace transform are unrelated operations.
• D) Convolution in time domain corresponds to multiplication in the Laplace domain.

If L{f(t)}=F(s), what is the Laplace transform of f(t−2)⋅u(t−2)?

• A) e2s⋅F(s)
• B) e-2s/ F(s)
• C) e-2s⋅F(s)
• D) F(s-2)

What is the Laplace transform of (t−3)2⋅u(t−3)?

• A) e-3s./s2
• B) es.2/s2
• C) e-3s.2/s3
• D) 2/s3

What is the Laplace transform of the second derivative f′′(t) of a function f(t)?

• A) s2F(s)+sf(0)+f′(0)
• B) s2F(s)−sf(0)−f′(0)
• C) s2F(s)+sf(0)−f′(0)
• D) s2F(s)−sf(0)+f′(0)

What is the Laplace transform of the first derivative f′(t) of a function f(t)?

• A) (1/s)F(s)+f(0)
• B) sF(s)−f(0)
• C) (1/s)F(s)−f(0)
• D) sF(s)+f(0)

Which of the following best describes the effect of unit step inputs u(t−1) and u(t−2) on the system response?

• A) They apply constant forces starting at t=1 and t=2.
• B) They turn the system on gradually over time.
• C) They act as impulse forces at t=1and t=2.
• D) They produce instantaneous jumps in velocity.

What is the Laplace transform of the Dirac delta function δ(t)?

• A) ∞
• B) 1
• C) 1/s
• D) 0

Which of the following is the correct Laplace transform of δ(t−3)?

• A) e3s
• B) s-3
• C) e3s /s
• D) e-3s

In solving the differential equation : y′′+3y′+2y=δ(t−1), y(0)=y′(0)=0 what is the Laplace transform of the right-hand side?

• A) 1/s
• B) δ(s−1)
• C) se-s
• D) e-s

Which of the following statements correctly defines the Dirac delta function δ(t)?

• A) It is infinite at t=0, zero elsewhere, and integrates to 1.
• B) It is zero for all t, including at t=0.
• C) It is a constant function with value 1 at t=0.
• D) It increases linearly from 0 to 1.

A voltage waveform starts at 0 V, jumps to 1 V at t = 1 s, to 2 V at t = 2 s, to 3 V at t = 3 s, and returns to 0 V at t = 4 s. Which of the following represents this signal using unit steps?

• A) u(t−1)+u(t−2)+u(t−3)−3u(t−4)
• B) u(t−1)+u(t−2)+u(t−3)
• C) u(t−1)−u(t−2)−u(t−3)+3 u(t−4)
• D) u(t−1)+2u(t−2)+3u(t−3)

Given the piecewise function: f(t) = \begin{cases} 0, & t < 2 \\ 5, & 2 \leq t < 4 \\ 0, & t \geq 4 \end{cases} Which of the following correctly represents f(t) using unit step functions?

• A) f(t)=5⋅u(t−2)
• B) f(t)=5⋅[u(t)−u(t−2)]
• C) f(t)=5⋅[u(t−2)+u(t−4)]
• D) f(t)=5⋅[u(t−2)−u(t−4)]

Given the piecewise function: \(f(t) = \begin{cases} 0, & t < 2 \\ 5, & 2 \leq t < 4 \\ 0, & t \geq 4 \end{cases} \) Which of the following correctly represents f(t) using unit step functions?

• A) f(t)=5⋅[u(t−2)+u(t−4)]
• B) f(t)=5⋅[u(t−2)−u(t−4)]
• C) f(t)=5⋅u(t−2)
• D) f(t)=5⋅[u(t)−u(t−2)]