The variance of a random variable X is $$\sigma _x^2\,.$$ Then the variance of - kx (where k is a positive constant ) is

A system with derivative control is shown in figure.
(a)Determine the transfer function and the characteristic equation of the system.

(b)Given that K > 0, find the range of '$$\tau $$' for which the system is unconditionally stable.

The magnitude plot of the open loop transfer function G(s), of a certain system is given in fig,

(a) Determine G(s), if it is known that the system is of minimum phase type.

(b) Estimate the phase at each of the corner frequencies.

The popular plot of G(s)=$${{10} \over {s{{\left( {s + 1} \right)}^2}}},$$ intercepts real axix at $$\omega = {\omega _0}$$ Then, the real part and $${\omega _0}$$ are respectively given by