An open loop control system results in a response of $${e^{ - 2t}}\left( {\sin 5t + \cos 5t} \right)$$ for a unit impulse input. The DC gain of the control system is __________.

The following discrete-time equations result from the numerical integration of the differential equations of an un-damped simple harmonic oscillator with state variables $$𝑥$$ and $$𝑦.$$ The integration time step is $$h.$$ $$${{{x_{k + 1}} - {x_k}} \over h} = {y_k},\,\,\,\,\,{{{y_{k + 1}} - {y_k}} \over h} = {x_k}$$$

For this discrete-time system, which one of the following statements is TRUE?

An open loop transfer function $$G(s)$$ of system is $$G\left( s \right) = {k \over {s\left( {s + 1} \right)\left( {s + 2} \right)}}$$

For a unity feedback system, the breakaway point of the root loci on the real axis occurs at,

Nyquist plots of two functions $${G_1}\left( s \right)$$ and $${G_2}\left( s \right)$$ are shown in figure.

Nyquist plot of the product of $${G_1}\left( s \right)$$ and $${G_2}\left( s \right)$$ is