The Nyquist plot of a system is given in the figure below. Let $\omega_{\mathrm{P}}, \omega_Q, \omega_R$, and $\omega_{\mathrm{S}}$ be the positive frequencies at the points $P, Q, R$, and $S$, respectively. Which one of the following statements is TRUE?

Consider the unity-negative-feedback system shown in Figure (i) below, where gain $K \geq 0$. The root locus of this system is shown in Figure (ii) below. For what value(s) of $K$ will the system in Figure (i) have a pole at $-1+j 1$ ?

Let $G(s)=\frac{1}{10 s^2}$ be the transfer function of a second-order system. A controller $M(s)$ is connected to the system $G(s)$ in the configuration shown below. Consider the following statements.

(i) There exists no controller of the form $M(s)=\frac{K_I}{s}$, where $K_I$ is a positive real number, such that the closed loop system is stable.

(ii) There exists at least one controller of the form $M(s)=K_P+s K_D$, where $K_P$ and $K_D$ are positive real numbers, such that the closed loop system is stable.

Which one of the following options is correct?

Consider the polynomial $p(s)=s^5+7 s^4+3 s^3-33 s^2+2 s-40$. Let $(L, I, R)$ be defined as follows. $L$ is the number of roots of $p(s)$ with negative real parts. $I$ is the number of roots of $p(s)$ that are purely imaginary. $R$ is the number of roots of $p(s)$ with positive real parts. Which one of the following options is correct?