# Objective Questions on Laplace Transform

## Objective Questions on Laplace Transform

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### Multiple Choice Questions

1. Laplace transform of a unit impulse function is
• S
• 0
• e-s
• 1

2. Laplace transforms of a damped sine wave e−αt sin (θt) · u(t) is

• $\frac{1}{(s+\alpha )^{2}+\phi ^{2}}$
• $\frac{s}{(s+\alpha )^{2}+\phi ^{2}}$
• $\frac{\phi ^{2}}{(s+\alpha )^{2}+\phi ^{2}}$
• $\frac{\phi}{(s+\alpha )^{2}+\phi ^{2}}$

Answer: $\frac{\phi ^{2}}{(s+\alpha )^{2}+\phi ^{2}}$

3. The final value of f(t) for a given $F(s)=\frac{s}{(s+4)(s+2)}$

• Zero
• 1/15
• 1/8
• 1/6

4. Laplace transforms of the function e−2t is

• $\frac{1}{2s}$
• (s + 2)
• $\frac{1}{s+2}$
• 2s

Answer: $\frac{1}{s+2}$

5. The Laplace transform of a function is $\frac{1}{s}Ee^{-as}$. The function is

• E sin ωt
• Eeat
• Eu(t − a)
• E cos ωt

6. If f(t) = r(t − α), f (s) =

• $\frac{e^{-as}}{s^{2}}$
• $\frac{\alpha }{s+\alpha }$
• $\frac{1}{s+\alpha }$
• $\frac{e^{-as}}{s}$

Answer: $\frac{e^{-as}}{s^{2}}$

7. The integral of a step function is

• A ramp function
• An impulse function
• Modified ramp function
• A sinusoidal function

8. Laplace transform of the function f (t) = (1 − e−αt) sin αt, where α is a constant is

• $\frac{s}{s^{2}+\alpha ^{2}}-\frac{s+\alpha }{(s+\alpha )^{2}+\alpha ^{2}}$
• $\frac{1}{s^{2}+\alpha ^{2}}-\frac{\alpha }{(s+\alpha )^{2}+\alpha ^{2}}$
• $\frac{\alpha }{s^{2}+\alpha ^{2}}-\frac{\alpha }{(s+\alpha )^{2}+\alpha ^{2}}$
• None of the above

Answer: $\frac{\alpha }{s^{2}+\alpha ^{2}}-\frac{\alpha }{(s+\alpha )^{2}+\alpha ^{2}}$

9. Laplace inverse equation $\frac{1}{(s+2)(s+3)}$ is

(a)e−t − e−2t

(b)e e−2t

(c)e e2t

(d)NOT

10. Laplace transform equation tn eat is

• $\frac{n!}{s^{n+1}}$
• $\frac{n!}{s^{n-1}}$
• $\frac{n!}{(s-a)^{n-1}}$
• $\frac{n!}{(s-a)^{n+1}}$

Answer: $\frac{n!}{(s-a)^{n-1}}$

11. Inverse Laplace transform for $\frac{2s+3}{s^{2}+3s}$ is

• 1 − e−3t
• 1 + e−3t
• 1 + e3t
• 1 − 3t

12. Inverse Laplace transform for $\frac{3s^{2}+4}{s(s^{2}+4)}$ is

• 1 + cos 2t
• 1 + 3cos 2t
• 1 + 2cos 2t
• 1 + 3cos t

13. Inverse Laplace transform for $\frac{6s}{(s+1)(s+2)(s+4)}$  is

• -2e−t + 6 e−2t + 4e−4t
• -2e−t + 6 e−2t − 4e−4t
• −2e−t + 6e2t + 4e−4t
• −2e−t + 6e−2t + 4e4t

Answer: -2e−t + 6 e−2t − 4e−4t

14. The value of function $f(s)=\frac{4(s+25)}{s(s+10)}$ at t = 0 is

• 10
• 4
• 0

15. Laplace transforms of tn u(t) is

• $\frac{n!}{s^{n}}$
• $\frac{n!}{s^{n-1}}$
• $\frac{(n-1)!}{s^{n-1}}$
• $\frac{n!}{s^{n+1}}$

Answer: $\frac{n!}{s^{n+1}}$

16. The final value of the function $I(s)=\frac{s+6}{s(s+3)}$ is

• 0
• 1
• 2
• 3

17. If $I(s)=\frac{3s}{(s+1)(s+4)}$, then i(t) is

• e−t + 4e-4t
• e−t + 4e4t
• −e−t + 4e−4t
• −e−t − 4e−4t

18. Laplace transform of t sin 2t is

• $\frac{4s}{(s^{2}+4)^{2}}$
• $\frac{s}{(s^{2}+4)^{2}}$
• $\frac{4s}{s^{2}+4}$
• $\frac{4s}{(s^{2}+4)^{3}}$

Answer: $\frac{4s}{(s^{2}+4)^{2}}$

19. If f1(t) = e −at and f2(t) = t, then convolution of f1(t) and f2(t) is

• $\frac{e^{at}}{a^{2}}\left [ ate^{at}+e^{at}+1 \right ]$
• $\frac{e^{-at}}{a^{2}}\left [ ate^{-at}-e^{at}+1 \right ]$
• $\frac{e^{-at}}{a^{2}}\left [ ate^{at}-e^{at}+1 \right ]$
• $\frac{e^{-at}}{a^{2}}\left [ ate^{at}-e^{at}-1 \right ]$

Answer: $\frac{e^{-at}}{a^{2}}\left [ ate^{at}-e^{at}+1 \right ]$

20. The Laplace transform of t cos 4t is

• $\frac{s^{2}-16}{(s^{2}+16)^{2}}$
• $\frac{s^{2}+16}{(s^{2}+16)^{2}}$
• $\frac{s^{2}}{(s^{2}+16)^{2}}$
• None of the above

Answer: $\frac{s^{2}-16}{(s^{2}+16)^{2}}$

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