MTH647 Quiz 3

Q1. Which of the following best describes the effect of unit step inputs u(t−1) and u(t−2) on the system response?

1. They produce instantaneous jumps in velocity.
2. They turn the system on gradually over time.
3. They apply constant forces starting at t=1 and t=2.
4. They act as impulse forces at t=1and t=2.

Correct answer: They apply constant forces starting at t=1 and t=2.

Q2. Given: f(t)=(t−3)2⋅u(t−3) What is the Laplace transform L{f(t)}?

1. 2/ s3
2. e−3s. 2/s3
3. 2e3s / s3
4. e-s s-2

Correct answer: e−3s. 2/s3

Q3. Which signal is represented by this diagram: zero for t < 0, then jumps to 1 at t = 0 and remains at 1?

1. Unit impulse
2. Unit parabola
3. Unit ramp
4. Unit step

Correct answer: Unit step

Q4. 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?

1. e-s
2. se-s
3. δ(s−1)
4. 1/s

Correct answer: e-s

Q5. Which of the following correctly represents the convolution of two continuous-time signals x(t) and h(t)?

1. y(t) = _{0}^{t} x(-1) \, h(1-) \, d
2. y(t) = _{-}^{} x() \, h() \, d
3. y(t) = _{0}^{t} x() \, h(t-) \, d
4. y(t) = _{0}^{} x() \, h(1-) \, d

Correct answer: y(t) = _{0}^{t} x() \, h(t-) \, d

Q6. 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?

1. u(t−1)+u(t−2)+u(t−3)−3u(t−4)
2. u(t−1)+u(t−2)+u(t−3)
3. u(t−1)−u(t−2)−u(t−3)+3 u(t−4)
4. u(t−1)+2u(t−2)+3u(t−3)

Correct answer: u(t−1)+u(t−2)+u(t−3)−3u(t−4)

Q7. Which of the following statements correctly defines the Dirac delta function δ(t)?

1. It is a constant function with value 1 at t=0.
2. It is zero for all t, including at t=0.
3. It is infinite at t=0, zero elsewhere, and integrates to 1.
4. It increases linearly from 0 to 1.

Correct answer: It is infinite at t=0, zero elsewhere, and integrates to 1.

Q8. Which of the following is the correct Laplace transform of δ(t−3)?

1. e3s /s
2. s-3
3. e3s
4. e-3s

Correct answer: e-3s

Q9. Given L{t2}=2/s3 , what is the Laplace transform of (t−2)2⋅u(t−2) using t-shifting theorem?

1. 2/s3. e2s
2. 2/(s3-2)
3. 2/s3. e-s
4. 2/s3. e-2s

Correct answer: 2/s3. e-2s

Q10. Which of the following correctly describes the relationship between the Dirac delta function δ(t) and the unit step function u(t)?

1. (t) = {d}{dt} u(t)
2. δ(t)=u(t)⋅t
3. \ u(t)= {d}{dt} (t)
4. u(t)=δ(t)⋅ t

Correct answer: δ(t)=u(t)⋅t