$A$ narrow parallel beam of light is incident paraxially on a solid transparent sphere of radius $r$ kept in air. What should be the refractive index if the beam is to be focused at the farther surface of the sphere?

When the temperature of a black body increases,it is observed that the wavelength corresponding to maximum energy changes from $0.26 \mu m$ to $0.13 \mu m$. The ratio of the emissive powers of the body at the respective temperatures is

The mass of oxygen gas occupying a volume of $11.2 \text{ litres}$ at a temperature $27^{\circ}C$ and a pressure of $760 \text{ mm}$ of mercury in kilograms is $[$Molecular weight of oxygen $= 32]$

$n$ conducting wires of same dimensions but having resistivities $1, 2, 3, . . . , n$ are connected in series. The equivalent resistivity of the combination is

The image formed by a convex mirror of focal length $30 \ cm$ is a quarter of the size of the object. The distance of the object from the mirror is $... \ cm$.

Two positively charged particles $m_{1}$ and $m_{2}$ have been accelerated across the same potential difference of $200 \ keV$ as shown below. Given mass of $m_{1} = 1 \ amu$ and $m_{2} = 4 \ amu$. The de Broglie wavelength of $m_{1}$ will be $x$ times that of $m_{2}$. The value of $x$ is . . . . . . . (nearest integer)