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 a system where $x_1(t), x_2(t)$, and $x_3(t)$ are three internal state signals and $u(t)$ is the input signal. The differential equations governing the system are given by
$$ \frac{d}{d t}\left[\begin{array}{l} x_1(t) \\ x_2(t) \\ x_3(t) \end{array}\right]=\left[\begin{array}{ccc} 2 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 0 \end{array}\right]\left[\begin{array}{l} x_1(t) \\ x_2(t) \\ x_3(t) \end{array}\right]+\left[\begin{array}{l} 1 \\ 1 \\ 1 \end{array}\right] u(t) $$
Which of the following statements is/are TRUE?