The GATE Grind

GATE 2023 IN – Question 37

Signals and Systems · Laplace and z-transforms · 2 marks · Multiple choice

The Laplace transform of $x(t)=e^{-3t}u(t-5)$, where $u$ is the unit step, is

  1. $e^{-5s}/(s+3),\quad\operatorname{Re}(s)>-3$
  2. $e^{-5(s-3)}/(s-3),\quad\operatorname{Re}(s)>3$
  3. $e^{-5(s+3)}/(s+3),\quad\operatorname{Re}(s)>-3$
  4. $e^{-5(s-3)}/(s+3),\quad\operatorname{Re}(s)>-3$

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (C) $e^{-5(s+3)}/(s+3),\quad\operatorname{Re}(s)>-3$

Explanation

Direct integration gives $X(s)=\int_5^\infty e^{-(s+3)t}\,dt=e^{-5(s+3)}/(s+3)$, converging for $\operatorname{Re}(s)>-3$. Answer: **C**. The exponential is not shifted with the step, so its amplitude at 5 s is already $e^{-15}$.