GATE 2016 EE (EE2) – Question 14

Signals and Systems · Linear time invariant and causal systems · 1 mark · Multiple choice

A causal LTI system satisfies $dy/dt+y/6=3x(t)$. Its zero-state response to $x(t)=3e^{-t/3}u(t)$ is

$9e^{-t/3}u(t)$
$9e^{-t/6}u(t)$
$(9e^{-t/3}-6e^{-t/6})u(t)$
$(54e^{-t/6}-54e^{-t/3})u(t)$

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Correct answer: (D) $(54e^{-t/6}-54e^{-t/3})u(t)$

Explanation

With zero initial state, $H(s)=3/(s+1/6)$ and $X(s)=3/(s+1/3)$. Hence $Y(s)=9/[(s+1/6)(s+1/3)]$.

Partial fractions give $Y=54/(s+1/6)-54/(s+1/3)$. Inverse transformation yields **$54(e^{-t/6}-e^{-t/3})u(t)$, D**. This also gives y(0+)=0 as required by the zero-state model.

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