The $$R-L-C$$ series circuit shown is supplied from a variable frequency voltage source. The admittance $$-$$ locus of the $$R-L$$ $$-C$$ network at terminals $$AB$$ for increasing frequency $$\omega $$ is

In the figure given below, all phasors are with reference to the potential at point $$''O''.$$ The locus of voltage phasor $${V_{YX}}$$ as $$R$$ is varied from zero to infinity is shown by

In the figure, transformer $${T_1}$$ has two secondaries, all three windings having the same number of turns and with polarities having the same number of turns and with polarities as indicated. One secondary is shorted by a $$10\,\Omega $$ resistor $$R,$$ and the other by a $$15\mu F$$ capacitor. The switch $$SW$$ is opened $$(t=0)$$ when the capacitor is charged to $$5$$ $$V$$ with the left plate as positive. At $$t=0+$$ the voltage $${V_P}$$ and current $${I_R}$$ are

A three phase balanced star connected voltage source with frequency $$\omega \,\,rad/s$$ is connected to a star connected balanced load which is purely inductive. The instantaneous line currents and phase to neutral voltages are denoted by $$\left( {{i_a},{i_b},{i_c}} \right)$$ and $$\left( {{V_{an}},\,\,{V_{bn}},\,\,{V_{cn}}} \right)$$ respectively and their $$rms$$ values are denoted by $$V$$ and $$1.$$ If $$$R = \left[ {{V_{an}}\,\,{V_{bn}}\,\,{V_{cn}}} \right]\left[ {\matrix{ 0 & {{1 \over {\sqrt 3 }}} & { - {1 \over {\sqrt 3 }}} \cr { - {1 \over {\sqrt 3 }}} & 0 & {{1 \over {\sqrt 3 }}} \cr {{1 \over {\sqrt 3 }}} & { - {1 \over {\sqrt 3 }}} & 0 \cr } } \right]\left[ {\matrix{ {{i_a}} \cr {{i_b}} \cr {{i_c}} \cr } } \right],$$$
then the magnitude of $$R$$ is