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 $$\mathrm{g}=10 \mathrm{~ms}^{-2}$$)

The amplitude of a particle executing SHM is $$3 \mathrm{~cm}$$. The displacement at which its kinetic energy will be $$25 \%$$ more than the potential energy is: __________ $$\mathrm{cm}$$

A certain pressure '$$\mathrm{P}$$' is applied to 1 litre of water and 2 litre of a liquid separately. Water gets compressed to $$0.01 \%$$ whereas the liquid gets compressed to $$0.03 \%$$. The ratio of Bulk modulus of water to that of the liquid is $$\frac{3}{x}$$. The value of $$x$$ is ____________.