Two circular coils $$P$$ and $$Q$$ of 100 turns each have same radius of $$\pi \mathrm{~cm}$$. The currents in $$P$$ and $$R$$ are $$1 A$$ and $$2 A$$ respectively. $$P$$ and $$Q$$ are placed with their planes mutually perpendicular with their centers coincide. The resultant magnetic field induction at the center of the coils is $$\sqrt{x} ~m T$$, where $$x=$$ __________.

[Use $$\mu_0=4 \pi \times 10^{-7} \mathrm{~TmA}^{-1}$$]

Two identical spheres each of mass $$2 \mathrm{~kg}$$ and radius $$50 \mathrm{~cm}$$ are fixed at the ends of a light rod so that the separation between the centers is $$150 \mathrm{~cm}$$. Then, moment of inertia of the system about an axis perpendicular to the rod and passing through its middle point is $$\frac{x}{20} \mathrm{~kg} \mathrm{m^{2 }}$$, where the value of $$x$$ is ___________.

Light from a point source in air falls on a convex curved surface of radius $$20 \mathrm{~cm}$$ and refractive index 1.5. If the source is located at $$100 \mathrm{~cm}$$ from the convex surface, the image will be formed at ________ $$\mathrm{cm}$$ from the object.

A nucleus has mass number $$A_1$$ and volume $$V_1$$. Another nucleus has mass number $$A_2$$ and Volume $$V_2$$. If relation between mass number is $$A_2=4 A_1$$, then $$\frac{V_2}{V_1}=$$ __________.