GATE 2022 IN Q.40: solution and answer

GATE 2022 · INQ.40 · 2 marks · MCQ

The signal $x (t) = (t - 1)^{2} u (t - 1)$ , where $u (t)$ is the unit-step function, has the Laplace transform $X (s)$ . The value of $X (1)$ is ________

(A)
$\frac{1}{e}$
(B)
$\frac{2}{e}$
(C)
$2 e$
(D)
$e^{2}$

See the question in the paper

Tap to open full size

Answer and solution

ANSWER

$X (s) = \frac{2 e ^{- s}}{s ^{3}}$ , so $X (1) = \frac{2}{e}$ . Option (B).

Official answer: BSource: Official answer key, GATE 2022 IN (IIT Kharagpur)

Write $x (t) = g (t - 1)$ with $g (t) = t^{2} u (t)$ . The signal is exactly $g (t)$ delayed by 1 s, because the squared term $(t - 1)^{2}$ and the step $u (t - 1)$ are both shifted by the same amount.
Standard pair: $L {t^{n} u (t)} = \frac{n !}{s ^{n + 1}}$ , so $G (s) = \frac{2 !}{s ^{3}} = \frac{2}{s ^{3}}$ , with ROC $Re (s) > 0$ .
Time-shifting property: $L {g (t - t_{0}) u (t - t_{0})} = e^{- s t_{0}} G (s)$ . With $t_{0} = 1$ : $X (s) = \frac{2 e ^{- s}}{s ^{3}}, Re (s) > 0$
$s = 1$ lies inside the ROC, so $X (1)$ exists. Substitute: $X (1) = \frac{2 e ^{- 1}}{1 ^{3}} = \frac{2}{e}$
Check by direct integration: $X (1) = \int_{1}^{\infty} (t - 1)^{2} e^{- t} d t$ . Put $τ = t - 1$ : $X (1) = e^{- 1} \int_{0}^{\infty} τ^{2} e^{- τ} d τ = e^{- 1} Γ (3) = \frac{2}{e}$ The answer is option (B).

KEY IDEA

The signal is $t^{2} u (t)$ shifted right by 1 s, with the squared term and the step shifted together. So you can use the time-shift property: $X (s) = e^{- s} \cdot \frac{2}{s ^{3}}$ . Then check that $s = 1$ is inside the ROC $Re (s) > 0$ before you substitute.

WATCH OUT FOR

Two mistakes are common. (1) Using $L {t^{2} u (t)} = 1/ s^{3}$ and dropping the $2!$ , which gives $X (1) = 1/ e$ , option (A). (2) Getting the sign of the shift exponent wrong ( $e^{+ s}$ in place of $e^{- s}$ ), which gives $2 e$ , option (C). The time-shift property applies directly only when the step is delayed by the same amount as the rest of the expression, as it is here.

From the GATE 2022 IN paper. Every question of this paper · Download the full paper · All 8 Laplace questions, with notes

GATE 2022 IN Q.40: solution and answer

Answer and solution

More Laplace questions