GATE Core Subject Knowledge & Technical Concepts 3 — Questions and Answers
Question 1: The Laplace transform of the unit step function u(t) is:
- 1/s²
- 1/s (Correct answer)
- s
- 1/(s+1)
Correct answer: 1/s
The Laplace transform of u(t) is 1/s for Re(s) > 0.
Question 2: A system is BIBO stable if and only if its impulse response h(t) satisfies:
- h(t) is bounded
- ∫|h(t)|dt < ∞ (Correct answer)
- h(t) → 0 as t → ∞
- h(t) is periodic
Correct answer: ∫|h(t)|dt < ∞
A system is BIBO stable if and only if its impulse response is absolutely integrable (L1 norm is finite).
Question 3: In a second-order control system, the damping ratio ζ = 0.5 corresponds to which response?
- Overdamped
- Critically damped
- Underdamped (Correct answer)
- Undamped
Correct answer: Underdamped
When 0 < ζ < 1 (here ζ = 0.5), the system is underdamped and exhibits oscillatory response.
Question 4: The Nyquist sampling theorem states that the sampling frequency must be:
- Equal to the signal's highest frequency
- At least twice the signal's highest frequency (Correct answer)
- Half the signal's highest frequency
- 10 times the signal's highest frequency
Correct answer: At least twice the signal's highest frequency
The sampling frequency must be at least 2fmax (Nyquist rate) to allow perfect reconstruction of the signal.
Question 5: For a causal LTI system, which condition must the region of convergence (ROC) of its z-transform satisfy?
- ROC includes the origin
- ROC is the entire z-plane
- ROC extends to infinity (includes |z| = ∞) (Correct answer)
- ROC is a ring in the z-plane
Correct answer: ROC extends to infinity (includes |z| = ∞)
For a causal (right-sided) sequence, the ROC is of the form |z| > r and must include |z| = ∞.
Question 6: The phase margin of a stable control system is defined as:
- The gain at the phase crossover frequency
- 180° + the phase of the open-loop transfer function at the gain crossover frequency (Correct answer)
- The frequency at which gain becomes 0 dB
- The difference between gain margin and unity
Correct answer: 180° + the phase of the open-loop transfer function at the gain crossover frequency
Phase margin = 180° + ∠G(jω) at the gain crossover frequency where |G(jω)| = 1 (0 dB).
Question 7: In AM modulation with modulation index m, the ratio of sideband power to total power is:
- m²/(2 + m²) (Correct answer)
- m/(2 + m)
- m²/2
- 2m²/(1 + m²)
Correct answer: m²/(2 + m²)
Total AM power = Pc(1 + m²/2); sideband power = Pc·m²/2, so ratio = (m²/2)/(1 + m²/2) = m²/(2 + m²).
GATE Core Subject Knowledge & Technical Concepts 3
The Laplace transform of the unit step function u(t) is: