A random variable $$X$$ has probability density function $$f(x)$$ as given below: $$$\,\,f\left( x \right) = \left\{ {\matrix{ {a + bx} & {for\,\,0 < x < 1} \cr 0 & {otherwise} \cr } } \right.\,\,$$$
If the expected value $$\,\,E\left[ X \right] = 2/3,\,\,$$ then $$\,\,\Pr \left[ {X < 0.5} \right]\,\,$$ is __________.

A solution of the ordinary differential equation $$\,\,{{{d^2}y} \over {d{t^2}}} + 5{{dy} \over {dt}} + 6y = 0\,\,$$ is such that $$y(0)=2$$ and $$y(1)=$$ $$ - \left( {{{1 - 3e} \over {{e^3}}}} \right).$$ The value of $${{dy} \over {dt}}\left( 0 \right)$$ is

In the following chopper, the duty ratio of switch $$S$$ is $$0.4.$$ If the inductor and capacitor are sufficiently large to ensure continuous inductor current and ripple free capacitor voltage, the charging current (in Ampere) of the $$5$$ $$V$$ battery, under steady-state, is_____

The circuit shown is meant to supply a resistive load $${R_L}$$ from two separate $$DC$$ voltage sources. The switches $$S1$$ and $$S2$$ are controlled so that only one of them is ON at any instant. $$S1$$ is turned on for $$0.2$$ $$ms$$ and $$S2$$ is turned on for $$0.3$$ $$ms$$ in a $$0.5$$ $$ms$$ switching cycle time period. Assuming continuous conduction of the inductor current and negligible ripple on the capacitor voltage, the output voltage $${V_0}$$ (in Volt) across $${R_L}$$ is __________.