A thin cylindrical rod of length $$10 \mathrm{~cm}$$ is placed horizontally on the principle axis of a concave mirror of focal length $$20 \mathrm{~cm}$$. The rod is placed in a such a way that mid point of the rod is at $$40 \mathrm{~cm}$$ from the pole of mirror. The length of the image formed by the mirror will be $$\frac{x}{3} \mathrm{~cm}$$. The value of $$x$$ is _____________.

A small particle moves to position $$5 \hat{i}-2 \hat{j}+\hat{k}$$ from its initial position $$2 \hat{i}+3 \hat{j}-4 \hat{k}$$ under the action of force $$5 \hat{i}+2 \hat{j}+7 \hat{k} \mathrm{~N}$$. The value of work done will be __________ J.

A series LCR circuit is connected to an ac source of $$220 \mathrm{~V}, 50 \mathrm{~Hz}$$. The circuit contain a resistance $$\mathrm{R}=100 ~\Omega$$ and an inductor of inductive reactance $$\mathrm{X}_{\mathrm{L}}=79.6 ~\Omega$$. The capacitance of the capacitor needed to maximize the average rate at which energy is supplied will be _________ $$\mu \mathrm{F}$$.

A solid cylinder is released from rest from the top of an inclined plane of inclination $$30^{\circ}$$ and length $$60 \mathrm{~cm}$$. If the cylinder rolls without slipping, its speed upon reaching the bottom of the inclined plane is __________ $$\mathrm{ms}^{-1}$$. (Given $$\mathrm{g}=10 \mathrm{~ms}^{-2}$$)