For the modulated signal $x(t)=m(t) \cos \left(2 \pi f_c t\right)$, the message signal $m(t)=4 \cos (1000 \pi t)$ and the carrier frequency $f_c$ is 1 MHz . The signal $x(t)$ is passed through a demodulator, as shown in figure below. The output $y(t)$ of the demodulator is

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