A $$3$$ $$V$$ $$dc$$ supply with an internal resistance of $$2$$ $$\Omega $$ supplies a passive non-linear resistance characterized by the relation $${V_{NL}} = {{\rm I}^2}{}_{NL}$$. The power dissipated in the non-linear resistance is

The matrix $$A$$ given below is the node incidence matrix of a network. The columns correspond to braches of the network while the rows correspond to nodes. Let
$$V = {\left[ {{v_1}\,\,{v_2}....{v_6}} \right]^T}$$ denote the vector of branches voltages while
$${\rm I} = {\left[ {{i_1}\,{i_2}....{i_6}} \right]^T}$$ that of branch currents. The vector $$E = {\left[ {{e_1}\,{e_2}\,\,{e_3}\,{e_4}} \right]^T}$$ denotes the vector of node voltages relative to a common ground. $$$A = \left[ {\matrix{ 1 & 1 & 1 & 0 & 0 & 0 \cr 0 & { - 1} & 0 & { - 1} & 1 & 0 \cr { - 1} & 0 & 0 & 0 & { - 1} & { - 1} \cr 0 & 0 & { - 1} & 1 & 0 & 1 \cr } } \right]$$$

Which of the following statements is true?

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

The resonant frequency for the given circuit will be