The state variable formulation of a system is given as
$$\left[ {\matrix{ {\mathop {{x_1}}\limits^ \bullet } \cr {\mathop {{x_2}}\limits^ \bullet } \cr } } \right] = \left[ {\matrix{ { - 2} & 0 \cr 0 & { - 1} \cr } } \right]\left[ {\matrix{ {{x_1}} \cr {{x_2}} \cr } } \right] + \left[ {\matrix{ 1 \cr 1 \cr } } \right]u,\,\,{x_1}\left( 0 \right) = 0,$$
$${x_2}\left( 0 \right) = 0$$ and $$y = \left[ {\matrix{ 1 & 0 \cr } } \right]\left[ {\matrix{ {{x_1}} \cr {{x_2}} \cr } } \right]$$

The system is

A bulb in staircase has two switches, one switch being at the ground floor and the other one at the first floor. The bulb can be turned $$ON$$ and also can be turned $$OFF$$ by any one of the switches irrespective of the state of the other switch. The logic of switching of the bulb resembles

The clock frequency applied to the digital circuit shown in the figure below is $$1$$ $$kHz.$$ If the initial state of the output $$Q$$ of the flip-flop is $$‘0’,$$ then the frequency of the output waveform $$Q$$ in $$kHz$$ is

Three capacitors C₁, C₂ and C₃ whose values are 10μF, 5μF and 2μF respectively, have breakdown voltages of 10V, 5V and 2V respectively. For the interconnection shown, the maximum safe voltage in Volts that can be applied across the combination and the corresponding total charge in μC stored in the effective capacitance across the terminals are respectively