Laplace Transform MCQ

1. Consider a linear time-invariant system with transfer function


If the input is cos(t) and the steady-state output is A cos (t+α), then the value of A is

  1. 0.90
  2. 0.707
  3. 0.84
  4. 0.50
Answer. b

2. The solution of the differential equation, for t > 0, y”(t) + 2y'(t) + y(t) = 0 with initial conditions y(0) = 0 and y'(0) = 1, is (u(t) denotes the unit step function),

  1. te-t u(t)
  2. (e-1 – te-t) u(t)
  3. (-e-t + te-t) u(t)
  4. e-t u(t)
Answer. a

3. The laplace transform of f(t) = e2t sin(5t) u(t) is

  1. \frac{1}{s^2-4s+29}
  2. \frac{5}{s^2+5}
  3. \frac{s-2}{s^2-4s+29}
  4. \frac{5}{s+5}
Answer. a

4. Which one of the following statements is not true for a continous time causal and stable LTI system?

  1. All the poles of the system must lie on the left side of the jω-axis.
  2. Zeros of the system can lie anywhere in the s-plane.
  3. All the poles must lie within |s| = 1.
  4. All the roots of the characteristics equation must be located on the left side of the jω-axis.
Answer. c

5. Consider a causal LTI system characterized by differential equation \frac{\mathrm{d} y(t)}{\mathrm{d} t}+\frac{1}{6}y(t)=3x(t). The response of the system to the input x(t)=3e^{-\frac{t}{3}}u(t), where u(t) denotes the unit step function, is

  1. 9e^{-\frac{t}{3}}u(t)
  2. 9e^{-\frac{t}{6}}u(t)
  3. 9e^{-\frac{t}{3}}u(t) - 6e^{-\frac{t}{6}}u(t)
  4. 54e^{-\frac{t}{6}}u(t) - 54e^{-\frac{t}{3}}u(t)
Answer. d

6. The transfer function of a system is \frac{Y(s)}{R(s)}=\frac{s}{s+2}. The steady state output y(t) is A cos(2t + Φ) for the input cos(2t). The values of A and Φ, respectively are?

  1. \frac{1}{\sqrt{2}},-45^{\circ}
  2. \frac{1}{\sqrt{2}},+45^{\circ}
  3. \sqrt{2},-45^{\circ}
  4. \sqrt{2},+45^{\circ}
Answer. b

7. The laplace transform of f(t)=2\sqrt{t/\pi } is s-3/2. The laplace transform of g(t)=\sqrt{1/\pi t } is

  1. \frac{3s^{-5/2}}{2}
  2. s-1/2
  3. s1/2
  4. s3/2
Answer. b

8. The unilateral laplace transform of f(t) is \frac{1}{s^2+s+1}. The unilateral laplace transform of t f(t) is

  1. -\frac{s}{(s^2+s+1)^2}
  2. -\frac{2s+1}{(s^2+s+1)^2}
  3. \frac{s}{(s^2+s+1)^2}
  4. \frac{2s+1}{(s^2+s+1)^2}
Answer. d

9. Let the laplace transform of a function f(t) which exists for t > 0 be Ft(s) and the laplace transform of its delayed version f(t-τ) be F2(s). F*1(s) be the complex conjugate of F1(s) with the laplace variable set as s = σ + jω. If G(s)=\frac{F_2(s)F_{1}^{*}(s)}{\left | F_1(s) \right |^2}, then the inverse laplace transform of G(s) is

  1. an ideal impulse δ(t)
  2. an ideal delayed impulse δ(t-τ)
  3. an ideal step function u(t)
  4. an ideal delayed step function u(t-τ)
Answer. b

10. The laplace transform of g(t) is

  1. \frac{1}{s}(e^{3s}-e^{5s})
  2. \frac{1}{s}(e^{-5s}-e^{-3s})
  3. \frac{e^{-3s}}{s}(1-e^{-2s})
  4. \frac{1}{s}(e^{5s}-e^{3s})
Answer. c

11. The laplace transform of (t2 – 2t) u(t-1) is

  1. \frac{2}{s^3}e^{-s}-\frac{2}{s^2}e^{-s}
  2. \frac{2}{s^3}e^{-2s}-\frac{2}{s^2}e^{-s}
  3. \frac{2}{s^3}e^{-s}-\frac{1}{s}e^{-s}
  4. None of the above
Answer. c
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