Properties of Laplace Transform MCQs With Answer

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Mastering Properties of Laplace Transform MCQs With Answer is essential for B.Pharm students dealing with pharmacokinetics, compartment models and systems analysis. This concise introduction covers core properties—linearity, time and frequency shifting, scaling, convolution, differentiation and integration in time and s-domain—plus initial and final value theorems and region of convergence. Understanding these properties helps convert time-domain drug concentration equations to s-domain algebraic forms for easier solution and interpretation of stability and transfer functions. These targeted MCQs emphasize application to drug modeling, ODE solutions, inverse transforms and practical table look-ups, strengthening analytic skills for exams and research. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. What is the Laplace transform definition for a causal time function f(t)?

  • F(s) = ∫_{-∞}^{∞} f(t) e^{-st} dt
  • F(s) = ∫_{0}^{∞} f(t) e^{-st} dt
  • F(s) = ∫_{0}^{T} f(t) e^{st} dt
  • F(s) = ∫_{-∞}^{0} f(t) e^{-st} dt

Correct Answer: F(s) = ∫_{0}^{∞} f(t) e^{-st} dt

Q2. Which property states L{a f(t) + b g(t)} = a F(s) + b G(s)?

  • Time shifting
  • Linearity
  • Convolution
  • Scaling

Correct Answer: Linearity

Q3. The Laplace transform of the Dirac delta δ(t) is:

  • 0
  • 1
  • 1/s
  • s

Correct Answer: 1

Q4. L{u(t)} where u(t) is the unit step function equals:

  • 1
  • 1/s
  • s
  • e^{-s}

Correct Answer: 1/s

Q5. Time shifting property: L{f(t – a) u(t – a)} equals:

  • e^{a s} F(s)
  • e^{-a s} F(s)
  • F(s – a)
  • F(s + a)

Correct Answer: e^{-a s} F(s)

Q6. Frequency shifting (multiplication by e^{at}): L{e^{a t} f(t)} equals:

  • F(s + a)
  • F(s – a)
  • e^{-a s} F(s)
  • e^{a s} F(s)

Correct Answer: F(s – a)

Q7. The Laplace transform of e^{a t} is:

  • 1/(s + a)
  • 1/(s – a)
  • s/(s^2 – a^2)
  • a/(s – a)

Correct Answer: 1/(s – a)

Q8. Convolution in time domain corresponds to which operation in s-domain?

  • Division of transforms
  • Addition of transforms
  • Multiplication of transforms
  • Convolution in s-domain

Correct Answer: Multiplication of transforms

Q9. L{f'(t)} for causal f(t) with initial value f(0+) equals:

  • s F(s)
  • s F(s) – f(0+)
  • F(s)/s
  • −dF/ds

Correct Answer: s F(s) – f(0+)

Q10. Multiplication by t in time domain corresponds to which s-domain operation?

  • −dF/ds
  • dF/ds
  • s F(s)
  • F(s)/s

Correct Answer: −dF/ds

Q11. L{t^n} (n integer ≥0) equals:

  • n! / s^{n+1}
  • s^{n} / n!
  • 1 / s^{n}
  • n / s^{n+1}

Correct Answer: n! / s^{n+1}

Q12. The initial value theorem states lim_{t→0+} f(t) = lim_{s→∞} ?

  • F(s)
  • s F(s)
  • F(s)/s
  • 1/F(s)

Correct Answer: s F(s)

Q13. The final value theorem holds if which condition on poles is satisfied?

  • All poles have positive real parts
  • No poles on the right-half plane and simple at origin allowed
  • All poles strictly in left-half plane except simple pole at origin is allowed
  • Poles on imaginary axis are allowed

Correct Answer: All poles strictly in left-half plane except simple pole at origin is allowed

Q14. L{cos ω t} equals:

  • s/(s^2 + ω^2)
  • ω/(s^2 + ω^2)
  • s/(s^2 – ω^2)
  • 1/(s^2 + ω^2)

Correct Answer: s/(s^2 + ω^2)

Q15. L{sin ω t} equals:

  • s/(s^2 + ω^2)
  • ω/(s^2 + ω^2)
  • ω/(s^2 – ω^2)
  • 1/(s + jω)

Correct Answer: ω/(s^2 + ω^2)

Q16. Time scaling: L{f(a t)} for a>0 equals:

  • (1/a) F(s/a)
  • a F(a s)
  • F(s/a)
  • (1/a) F(a s)

Correct Answer: (1/a) F(s/a)

Q17. Division by t in time domain (integral) corresponds to which s-domain relation for causal f(t)?

  • F(s)/s
  • s F(s)
  • −dF/ds
  • Integral of F(s) ds

Correct Answer: F(s)/s

Q18. L{H(t-a)} where H is unit step shifted by a equals:

  • e^{-a s}/s
  • e^{a s}/s
  • 1/(s-a)
  • e^{-a s}

Correct Answer: e^{-a s}/s

Q19. For a one-compartment first-order elimination model C(t)=C0 e^{-k t}, the Laplace transform is:

  • C0/(s + k)
  • C0/(s – k)
  • C0 s/(s + k)
  • C0 k/(s + k)

Correct Answer: C0/(s + k)

Q20. Bilateral Laplace transform differs from unilateral by:

  • Integration from 0 to ∞ instead of −∞ to ∞
  • Including integration from −∞ to ∞
  • Multiplication by e^{st}
  • Excluding the ROC

Correct Answer: Including integration from −∞ to ∞

Q21. The region of convergence (ROC) is important because it:

  • Determines existence and causality of transform
  • Only gives inverse transform numerically
  • Is always all complex plane
  • Is irrelevant to stability

Correct Answer: Determines existence and causality of transform

Q22. Multiplication by e^{-a t} in time domain corresponds to which s-domain shift?

  • F(s – a)
  • F(s + a)
  • e^{-a s} F(s)
  • (1/a) F(s/a)

Correct Answer: F(s + a)

Q23. L^{-1}{1/(s – a)} equals:

  • e^{a t}
  • e^{-a t}
  • δ(t – a)
  • u(t) e^{a t}

Correct Answer: e^{a t}

Q24. If F(s) has a simple pole at s = −α (α>0), the time-domain term is:

  • e^{α t}
  • e^{-α t}
  • t e^{-α t}
  • sin(α t)

Correct Answer: e^{-α t}

Q25. Partial fraction decomposition is primarily used to:

  • Solve ODEs numerically
  • Perform inverse Laplace transforms analytically
  • Find ROC boundaries
  • Compute unilateral transforms only

Correct Answer: Perform inverse Laplace transforms analytically

Q26. Which property helps solve linear ODEs with initial conditions using Laplace transforms?

  • Convolution theorem
  • Differentiation property (s-domain algebraic conversion)
  • Time scaling
  • Frequency modulation

Correct Answer: Differentiation property (s-domain algebraic conversion)

Q27. Laplace transform of a convolution f * g equals:

  • F(s) + G(s)
  • F(s) G(s)
  • F(s) / G(s)
  • G(s) – F(s)

Correct Answer: F(s) G(s)

Q28. L{∫_0^t f(τ) dτ} equals:

  • F(s) s
  • F(s)/s
  • −dF/ds
  • F(s + 1)

Correct Answer: F(s)/s

Q29. Multiplying F(s) by s in the s-domain corresponds to which time-domain operation (ignoring initial conditions)?

  • t f(t)
  • f'(t)
  • ∫ f(t) dt
  • f(-t)

Correct Answer: f'(t)

Q30. The transform pair L{e^{-k t} u(t)} and its ROC (k>0) is:

  • 1/(s + k), Re(s) > −k
  • 1/(s + k), Re(s) > k
  • 1/(s – k), Re(s) > k
  • 1/(s – k), Re(s) < k

Correct Answer: 1/(s + k), Re(s) > −k

Q31. For stability of a linear system represented by Laplace transform, where must poles lie?

  • Right-half s-plane
  • Left-half s-plane
  • On the imaginary axis
  • Anywhere except origin

Correct Answer: Left-half s-plane

Q32. The Laplace transform of a periodic function can be obtained using:

  • Time differentiation
  • A geometric series formula based on one period
  • Multiplication by t
  • Time reversal

Correct Answer: A geometric series formula based on one period

Q33. L{t e^{-a t}} equals:

  • 1/(s + a)^2
  • 1/(s – a)^2
  • 1/(s + a)
  • e^{-a s}/s^2

Correct Answer: 1/(s + a)^2

Q34. Differentiation of F(s) with respect to s corresponds to which time-domain multiplication?

  • Multiplication by t (positive)
  • Multiplication by −t
  • Time shift
  • Division by t

Correct Answer: Multiplication by −t

Q35. If input is an impulse δ(t) into a linear system with transfer function H(s), the output in s-domain is:

  • H(s) · δ(s)
  • H(s)
  • 1/H(s)
  • Convolution of H and δ

Correct Answer: H(s)

Q36. In pharmacokinetics, representing infusion input as constant rate r for t≥0 has Laplace transform:

  • r/s
  • r
  • r e^{-s}/s
  • r s

Correct Answer: r/s

Q37. The effect of multiplying f(t) by cos(ω0 t) corresponds to which s-domain operation (modulation)?

  • Average of F(s + jω0) and F(s − jω0)
  • Product F(s) cos(ω0)
  • Shift F(s ± ω0)
  • Convolution in s-domain

Correct Answer: Average of F(s + jω0) and F(s − jω0)

Q38. L{u(t − a)} where u is input pulse delayed a and represented as step equals:

  • e^{a s}/s
  • e^{-a s}/s
  • 1/(s − a)
  • e^{-a s}

Correct Answer: e^{-a s}/s

Q39. For inversion using residue theorem, residues are computed at:

  • Zeros of F(s)
  • Poles of F(s)·e^{s t}
  • Poles of F(s)
  • All points in ROC

Correct Answer: Poles of F(s)·e^{s t}

Q40. The Laplace transform helps convert differential equations into:

  • Higher-order differential equations
  • Integral equations only
  • Algebraic equations in s-domain
  • Nonlinear algebraic equations always

Correct Answer: Algebraic equations in s-domain

Q41. L{f(t)/t} cannot be obtained directly by a simple standard property; which approach is commonly used?

  • Use convolution theorem
  • Use integration in s-domain from F(s) to ∞
  • Use differentiation property
  • Time reversal

Correct Answer: Use integration in s-domain from F(s) to ∞

Q42. For L{f(t)} to exist, f(t) must be of exponential order, meaning:

  • |f(t)| ≤ Ce^{α t} for some C, α as t→∞
  • f(t) must be periodic
  • f(t) must be bounded only on finite interval
  • f(t) must be differentiable infinitely

Correct Answer: |f(t)| ≤ Ce^{α t} for some C, α as t→∞

Q43. If F(s) = 1/(s (s + a)), inverse Laplace corresponds to which time function?

  • 1/a (1 − e^{-a t})
  • e^{-a t}/a
  • t e^{-a t}
  • 1/(s + a)

Correct Answer: 1/a (1 − e^{-a t})

Q44. L{f(t) cosh(a t)} can be expressed in s-domain using shifts at:

  • s ± a
  • s ± j a
  • Only s + a
  • Only s − a

Correct Answer: s ± a

Q45. The Laplace transform of a convolution integral representing drug absorption convolution g * h gives the plasma response as:

  • G(s) + H(s)
  • G(s) H(s)
  • G(s) / H(s)
  • H(s) − G(s)

Correct Answer: G(s) H(s)

Q46. Which formula gives L{t^n e^{-a t}}?

  • n! / (s + a)^{n+1}
  • (s + a)^{n+1} / n!
  • n / (s − a)^{n+1}
  • n! / (s − a)^{n+1}

Correct Answer: n! / (s + a)^{n+1}

Q47. Multiplication by u(t) is implicit in unilateral Laplace; this ensures transform handles which type of signals?

  • Anti-causal signals only
  • Causal signals (t≥0)
  • Periodic signals only
  • Signals without initial conditions

Correct Answer: Causal signals (t≥0)

Q48. The relation L{f(t) * δ(t − a)} equals:

  • F(s) e^{−a s}
  • F(s + a)
  • F(s) + e^{−a s}
  • F(s) shifted in time domain

Correct Answer: F(s) e^{−a s}

Q49. For inverse Laplace by table lookup, the common decomposition step used is:

  • Fourier series expansion
  • Partial fraction decomposition
  • Laplace differentiation
  • Time-shift multiplication

Correct Answer: Partial fraction decomposition

Q50. In pharmacokinetic compartment models, transfer functions in s-domain represent:

  • Time-domain concentration directly
  • Algebraic relation between input and output processes (e.g., infusion to plasma concentration)
  • Only elimination rate constants numerically
  • Discrete sampling times only

Correct Answer: Algebraic relation between input and output processes (e.g., infusion to plasma concentration)

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