GATE 2017 EC (EC2) – Question 18
The input $x(t)$ and output $y(t)$ of a continuous-time system are related by $y(t)=\int_{t-T}^{t}x(u)\,du$. The system is
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Correct answer: (B) linear and time-invariant
Explanation
Integration obeys superposition: replacing the input by $ax_1+bx_2$ produces $ay_1+by_2$, so the system is linear.
For a shifted input $x(t-t_0)$, substitute $v=u-t_0$ in the integral. Its limits become $t-t_0-T$ and $t-t_0$, giving the original output evaluated at $t-t_0$.
Thus a time shift of the input produces the same shift of the output, for fixed T. The system is **linear and time-invariant, B**.