In L.P.P., the maximum value of objective function $\mathrm{Z}=6 x+3 y$ subject to constraints $x+y \leq 5, x+2 y \geq 4,4 x+y \leq 12, x, y \geq 0$ is

If the lines $x=a y-1=z-2$ and $x=3 y-2=\mathrm{bz}-2(\mathrm{ab} \neq 0)$ are coplanar, then

If $\left(\tan ^{-1} x\right)^2+\left(\cot ^{-1} x\right)^2=\frac{5 \pi^2}{8}$, then $x^2+1=$

If $p \equiv$ The switch $S_1$ is closed, $q \equiv$ The switch $\mathrm{S}_2$ is closed, $\mathrm{r} \equiv$ switch $\mathrm{S}_3$ is closed, then symbolic form of following switching circuit is equivalent to