Allied & ICAR Exam Bank • ALLIED_1022 MCQ • 2 Marks

Section 1: Engineering Mathematics

Differential Equations

Subtopic: Laplace transforms

Question Statement:

The Laplace transform of $f(t) = t e^{3t} \cos(2t)$ is:

Options:

  • (A) \frac{(s-3)^2 - 4}{[(s-3)^2 + 4]^2}
  • (B) \frac{2(s-3)}{[(s-3)^2 + 4]^2}
  • (C) \frac{(s-3)^2 + 4}{[(s-3)^2 - 4]^2}
  • (D) \frac{s-3}{[(s-3)^2 + 4]^2}

Answer Key:

A

Step-by-Step Solution & Explanation

$\mathcal{L}\{\cos(2t)\} = \frac{s}{s^2 + 4}$. Multiplication by $t$ gives $-\frac{d}{ds}\left(\frac{s}{s^2+4}\right) = -\frac{(s^2+4)(1) - s(2s)}{(s^2+4)^2} = \frac{s^2 - 4}{(s^2+4)^2}$. By shifting theorem $e^{3t}$, replace $s$ with $(s-3)$: $F(s) = \frac{(s-3)^2 - 4}{[(s-3)^2 + 4]^2}$.
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