GATE IN 2023 Question Paper (11-Feb-2023) (Shift-1)

A system has the transfer-function
$\frac{Y(s)}{X(s)}=\frac{s-\pi }{s+\pi }$.
Let $u(t)$ be the unit-step function. The input $x(t)$ that results in a steady-state output $y(t)=\sin \pi t$ is ______

Choose the fastest logic family among the following:

What is ${\lim }_{x\to 0}f(x)$, where $f(x)=x\sin \frac{1}{x}$?

The number of zeros of the polynomial $P(s)=s^3+2s^2+5s+80$ in the right-half plane is ______.

The number of times the Nyquist plot of $G(s)H(s)=\frac{1}{2}\frac{(s-1)(s-2)}{(s+1)(s+2)}$ encircles the origin is ______.