The number of independent loops for a network with n nodes and b branches is

Refer to the circuit shown in Fig. Choosing the voltage v_C(t) across capacitor, and the current i_L(t) through the inductor as state variable,i.e., $$$\left[\mathrm x\left(\mathrm t\right)\right]\;=\;\begin{bmatrix}{\mathrm v}_\mathrm C\left(\mathrm t\right)\\{\mathrm i}_\mathrm L\left(\mathrm t\right)\end{bmatrix}$$$ Write the state equation in the form $$\frac{\operatorname d\left[x\left(t\right)\right]}{\operatorname dt}\;=\;\left[A\right]\left[x\left(t\right)\right]\;+\;\left[B\right]\left[u\left(t\right)\right]$$ and find [A], [B] and [u(t)].

In the circuit shown in Fig (a) - (c), assuming initial voltages across capacitors and current through the inductors to be zero at the time of switching (t=0), then at any time t>0.

Match each of the items of Set 1, with the appropriate item of the Set 2.

Set 2

(1) current increases monotonically with time
(2) current decreases monotonically with time
(3) current remains constant at V/R
(4) current first increases, then decreases
(5) no current can ever flow

In the circuit shown in Fig., it is known that the variable current source I absorbs power.Find I (in magnitude and direction) so that it receives maximum power and also find the amount of power absorbed by it.