A second order system starts with an initial condition of $$\left( {\matrix{ 2 \cr 3 \cr } } \right)$$ without any external input. The state transition matrix for the system is given by $$\left( {\matrix{ {{e^{ - 2t}}} & 0 \cr 0 & {{e^{ - t}}} \cr } } \right).$$ The state of the system at the end of $$1$$ second is given by.

The following equation defines a separately exited $$dc$$ motor in the form of a differential equation $${{{d^2}\omega } \over {d{t^2}}} + {{B\,d\omega } \over {j\,\,dt}} + {{{K^2}} \over {LJ}}\omega = {K \over {LJ}}{V_a}$$

The above equation may be organized in the state space form as follows
$$\left( {\matrix{ {{{{d^2}\omega } \over {d{t^2}}}} \cr {{{d\omega } \over {dt}}} \cr } } \right) = P\left( {\matrix{ {{{d\omega } \over {dt}}} \cr \omega \cr } } \right) + Q{V_a}$$

where the $$P$$ matrix is given by

The Boolean expression $$X\overline Y Z + XYZ + \overline X Y\overline Z + \overline X \overline Y Z + XY\overline Z $$ can be simplified to

Figure shows a $$4$$ to $$1$$ $$MUX$$ to be used to implement the sum $$S$$ of a $$1$$-bit full adder with input bits $$P$$ and $$Q$$ and the carry input $${C_{in}}.$$ Which of the following combinations of inputs to $${{\rm I}_0},\,\,{{\rm I}_1},\,\,{{\rm I}_2}$$ and $$\,\,{{\rm I}_3}$$ of the $$MUX$$ will realize the sum $$S$$?