Laplace Transform MCQ

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

$H(s)=\frac{1}{(s+1)}$

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

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)

3. The laplace transform of f(t) = e^2t 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. $\frac{5}{s+5}$

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.

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. $54e^{-\frac{t}{6}}u(t) – 54e^{-\frac{t}{3}}u(t)$

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. $\sqrt{2},+45^{\circ}$

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. s^1/2
4. s^3/2

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. $\frac{2s+1}{(s^2+s+1)^2}$

9. Let the laplace transform of a function f(t) which exists for t > 0 be F_t(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)

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. $\frac{1}{s}(e^{5s}-e^{3s})$

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