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Laplace transforms and Fourier series: board exam questions with solutions

On the ECE exam under Mathematics and the REE exam under Mathematics. Try each question before you open its solution, then practise with fresh numbers below: 1 computed question type on this topic.

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Questions

1. What is the Laplace transform of e^(−at)?

  1. 1/(s − a)
  2. a/(s + a)
  3. s/(s + a)
  4. 1/(s + a)
Show the answer and solution

Answer: D. 1/(s + a)

∫₀^∞ e^(−at)e^(−st) dt = 1/(s + a), for s > −a.

2. What is the Laplace transform of sin ωt?

  1. s/(s² + ω²)
  2. ω/(s² − ω²)
  3. 1/(s² + ω²)
  4. ω/(s² + ω²)
Show the answer and solution

Answer: D. ω/(s² + ω²)

From the standard table, L{sin ωt} = ω/(s² + ω²). The s/(s² + ω²) form is for cos ωt.

3. Find the inverse Laplace transform of 5/[s(s + 5)].

  1. 5 − 5e^(−5t)
  2. 1 − e^(−5t)
  3. e^(−5t)
  4. 5e^(−5t)
Show the answer and solution

Answer: B. 1 − e^(−5t)

Partial fractions: 5/[s(s + 5)] = 1/s − 1/(s + 5), which inverts to 1 − e^(−5t).

4. Use the final value theorem to find the final value of f(t) when F(s) = 10/[s(s + 2)].

  1. 10
  2. 0
  3. 5
  4. 20
Show the answer and solution

Answer: C. 5

All poles of sF(s) are in the left half-plane, so f(∞) = lim(s→0) sF(s) = 10/2 = 5.

5. What is the Laplace transform of t²?

  1. 2/s³
  2. 1/s³
  3. 2/s²
  4. t²/s
Show the answer and solution

Answer: A. 2/s³

L{tⁿ} = n!/s^(n+1), so L{t²} = 2/s³.

6. An odd, half-wave symmetric square wave is expanded in a Fourier series. Which terms appear?

  1. Sine terms at odd harmonics only
  2. Cosine terms at all harmonics
  3. A DC term plus even harmonics
  4. Sine terms at even harmonics only
Show the answer and solution

Answer: A. Sine terms at odd harmonics only

Odd symmetry removes the DC and cosine terms. Half-wave symmetry removes the even harmonics. Only odd-harmonic sine terms remain (4A/πn for odd n).

7. Use the initial value theorem to find f(0⁺) when F(s) = (3s + 2)/(s² + 4s + 5).

  1. 0.4
  2. 2
  3. 3
  4. 0
Show the answer and solution

Answer: C. 3

f(0⁺) = lim(s→∞) sF(s) = lim (3s² + 2s)/(s² + 4s + 5) = 3. 0.4 is what the final-value limit at s → 0 would give.

8. Find the inverse Laplace transform of (s + 3)/[(s + 1)(s + 2)].

  1. e^(−t) + 2e^(−2t)
  2. 2e^(−t) − e^(−2t)
  3. 2e^(−t) + e^(−2t)
  4. 3e^(−t) − 2e^(−2t)
Show the answer and solution

Answer: B. 2e^(−t) − e^(−2t)

Write (s + 3)/[(s + 1)(s + 2)] = A/(s + 1) + B/(s + 2). A = (−1 + 3)/(−1 + 2) = 2 and B = (−2 + 3)/(−2 + 1) = −1, so f(t) = 2e^(−t) − e^(−2t).

Worked examples with computed numbers

These numbers are fixed on this page; in practice they change every time.

9. F(s) = 22/[s(s + 7)]. What is f(t) as t → ∞?

  1. 3.143
  2. 154
  3. 0
  4. 22
Show the answer and solution

Answer: A. 3.143

Final value theorem: lim s·F(s) as s → 0 = 22/(0 + 7) = 3.143.

Other topics in the same subjects

Algebra · Trigonometry · Analytic geometry · Differential calculus · Integral calculus · Differential equations · Complex numbers · Probability and statistics · Matrices and determinants

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