For the balanced 3-phase transmission line shown, consider the following cases:

Case-1: $\left|V_1\right|=1.1$ p.u., $\left|V_2\right|=0.9$ p.u., $Z=0.75 \angle 0^{\circ}$ p.u. and $\theta_{12}=\theta_1-\theta_2=0^{\circ}$

Case-2 : $\left|V_1\right|=1.1$ p.u., $\left|V_2\right|=0.9$ p.u., $Z=0.75 \angle 90^{\circ}$ p.u. and $\theta_{12}=\theta_1-\theta_2=90^{\circ}$

Which of the following statements is/are correct about real power loss and reactive power loss in the line?

A system with two generators G1 and G2 (without generator limits) is shown.

The total load on the system is 1184 MW . The expression for the cost of generation ( $\mathrm{C}_1$ and $\mathrm{C}_2$ ) and real power loss ( $P_{\text {loss }}$ ) are as follows:

$$ \begin{aligned} & \mathrm{C}_1\left(P_{G 1}\right)=1000+50 P_{G 1}+0.01\left(P_{G 1}\right)^2 \mathrm{Rs} / \mathrm{MWh} \\ & \mathrm{C}_2\left(P_{G 2}\right)=2000+50 P_{G 2}+0.001\left(P_{G 1}\right)^2 \mathrm{Rs} / \mathrm{MWh} \end{aligned} $$

$$ P_{\text {Loss }}=0.001\left(P_{G 2}-50\right)^2 \mathrm{MW} $$

When the generators are operating at their optimal generation, meeting the total load requirement, the real power loss in the system is $\_\_\_\_$ MW (Round off to one decimal place)

Consider the Lagrange multiplier $\lambda=70.25$ for optimal generation.

Consider the infinite-length, discrete-time sequence $x[n]=0.9^{|n|}$, where $n$ is an integer. The region of convergence of its Z-transform $X(z)$ is given by:

(Note: $z$ is a complex variable)

Let $x_C(t)$ be any continuous-time periodic signal with period $T$. It is sampled uniformly with a sampling period $T_s$ where $T_s \neq T$, resulting in the discrete sequence $x[n]=x_c\left(n T_s\right)$, where $n$ is an integer. Which one of the following statements is correct about $x[n]$ ?