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

Answer: 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

Answer: Zero

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
• Ee^at
• Eu(t − a)
• E cos ωt

Answer: Eu(t − a)

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

Answer: A ramp 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⁻ e^2t

(d)NOT

10. Laplace transform equation tⁿ e^at 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 + e^3t
• 1 − e ^3t

Answer: 1 + e^−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

Answer: 1 + 2cos 2t

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 + 6e^2t + 4e^−4t
• −2e^−t + 6e^−2t + 4e^4t

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
• ∞

Answer: 4

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

Answer: 2

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

• e^−t + 4e^-4t
• e^−t + 4e^4t
• −e^−t + 4e^−4t
• −e^−t − 4e^−4t

Answer: −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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