Question Bank • QB_EM_110 MCQ • 1 Mark

Section 1: Engineering Mathematics

Differential Equations

Subtopic: Laplace transforms and their inverse

Question Statement:

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

Options:

  • (A) $\frac{2}{(s - 3)^3}$
  • (B) $\frac{1}{(s - 3)^3}$
  • (C) $\frac{2}{(s + 3)^3}$
  • (D) $\frac{6}{(s - 3)^2}$

Answer Key:

A

Step-by-Step Solution & Explanation

By the first shifting theorem, $\mathcal{L}\{e^{at} f(t)\} = F(s - a)$. Since $\mathcal{L}\{t^2\} = \frac{2!}{s^3} = \frac{2}{s^3}$, shifting with $a = 3$ yields: $\mathcal{L}\{t^2 e^{3t}\} = \frac{2}{(s - 3)^3}$
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