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

$$ \text { Consider the following closed loop control system } $$

where $G(s)=\frac{1}{s(s+1)}$ and $C(s)=K \frac{s+1}{s+3}$. If the steady state error for a unit ramp input is 0.1 , then the value of $K$ is $\_\_\_\_$ .

The figure below shows a multiplexer where $S_1$ and $S_0$ are the select lines, $I_0$ to $I_3$ are the input data lines, $E N$ is the enable line, and $F(P, Q, R)$ is the output. $F$ is