The random variable $X$ takes values in $\{-1,0,1\}$ with probabilities $P(X=-1)=P(X=1)$ and $\alpha$ and $P(X=0)=1-2 \alpha$, where $0<\alpha<\frac{1}{2}$.

Let $g(\alpha)$ denote the entropy of $X$ (in bits), parameterized by $\alpha$. Which of the following statements is/are TRUE?

$X$ and $Y$ are Bernoulli random variables taking values in $\{0,1\}$. The joint probability mass function of the random variables is given by:

$$ \begin{aligned} & P(X=0, Y=0)=0.06 \\ & P(X=0, Y=1)=0.14 \\ & P(X=1, Y=0)=0.24 \\ & P(X=1, Y=1)=0.56 \end{aligned} $$

The mutual information $I(X ; Y)$ is ___________(rounded off to two decimal places).

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 $-1+j 1$ ?