The pole-zero map of a rational function $G(s)$ is shown below. When the closed contour $\Gamma$ is mapped into $G(s)$-plane, then the mapping encircles

The loop transfer function of a negative feedback system is

$$ G(s) H(s)=\frac{K(s+11)}{s(s+2)(s+8)} $$

The value of $K$, for which system is marginally stable, is $\_\_\_\_$ .

The characteristic equation of a system is

$$ s^3+3 s^2+(K+2) s+3 K=0 $$

In the root locus plot for the given system, as $K$ varies from 0 to $\infty$, the break-away or break-in point(s) lie within

A system with transfer function $G(s)=\frac{1}{(s+1)(s+a)}, a>0$ is subjected to input $5 \cos 3 t$. The steady state output of the system is $\frac{1}{\sqrt{10}} \cos (3 t-1.892)$. The value of $a$ is