A 50 Hz, 275 kV line of length 400 km has the following parameters:

Resistance, R = 0.035 $$\Omega$$/km;

Inductance, L = 1 mH/km;

Capacitance, C = 0.01 $$\mu$$F/km;

The line is represented by the nominal-$$\pi$$ model. With the magnitudes of the sending end and the receiving end voltages of the line (denoted by $$V_S$$ and $$V_R$$, respectively) maintained at 275 kV, the phase angle difference ($$\theta$$) between $$V_S$$ and $$V_R$$ required for maximum possible active power to be delivered to the receiving end, in degree is ___________ (Round off to 2 decimal places).

The geometric mean radius of a conductor, having four equal strands with each strand of radius r, as shown in the figure below, is

In the figure, the voltages are

$${v_1}\left( t \right) = 100\cos \left( {\omega t} \right)$$

$${v_2}\left( t \right) = 100\cos \left( {\omega t + {\pi \over {18}}} \right)$$

and $${v_3}\left( t \right) = 100\cos \left( {\omega t + {\pi \over {36}}} \right)$$.

The circuit is in sinusoidal steady state, and R << $${\omega L}$$. P₁, P₂ and P₃ are the average power outputs. Which one of the following statements is true?

The normal - $$\pi $$ circuit of a transmission line is shown in the figure.

Impedance $$Z = 100\angle {80^ \circ }$$ and reactance $$\,X = 3300\Omega .$$ The magnitude of the characteristic impedance of the transmission line, in $$\Omega ,$$ is ________. (Give the answer up to one decimal place.)