The gain margin of a unity feedback control system with the open loop transfer function $$G\left( s \right) = {{\left( {s + 1} \right)} \over {{s^2}}}$$ is

A system with zero initial conditions has the closed loop transfer function $$T\left( s \right) = {{{s^2} + 4} \over {\left( {s + 1} \right)\left( {s + 4} \right)}}.$$ The system output is zero at the frequency

In the $$GH(s)$$ plane, the Nyquist plot of the loop transfer function $$G\left( s \right)\,H\left( s \right) = {{\pi {e^{ - 0.25s}}} \over s}$$ passes through the negative real axis at the point

If the compensated system shown in the figure has a phase margin of $${60^ \circ }$$ at the crossover frequency of $$1 rad/sec,$$ the value of the gain $$K$$ is