Light enters from air into a given medium at an angle of $$45^{\circ}$$ with interface of the air-medium surface. After refraction, the light ray is deviated through an angle of $$15^{\circ}$$ from its original direction. The refractive index of the medium is:

A tube of length $$50 \mathrm{~cm}$$ is filled completely with an incompressible liquid of mass $$250 \mathrm{~g}$$ and closed at both ends. The tube is then rotated in horizontal plane about one of its ends with a uniform angular velocity $$x \sqrt{F} \,\mathrm{rad} \,\mathrm{s}^{-1}$$. If $$\mathrm{F}$$ be the force exerted by the liquid at the other end then the value of $$x$$ will be __________.

Nearly 10% of the power of a $$110 \mathrm{~W}$$ light bulb is converted to visible radiation. The change in average intensities of visible radiation, at a distance of $$1 \mathrm{~m}$$ from the bulb to a distance of $$5 \mathrm{~m}$$ is $$a \times 10^{-2} \mathrm{~W} / \mathrm{m}^{2}$$. The value of 'a' will be _________.

A metal wire of length $$0.5 \mathrm{~m}$$ and cross-sectional area $$10^{-4} \mathrm{~m}^{2}$$ has breaking stress $$5 \times 10^{8} \,\mathrm{Nm}^{-2}$$. A block of $$10 \mathrm{~kg}$$ is attached at one end of the string and is rotating in a horizontal circle. The maximum linear velocity of block will be _________ $$\mathrm{ms}^{-1}$$.