A parallel beam of light is allowed to fall on a transparent spherical globe of diameter $30 \mathrm{cm}$ and refractive index $1.5$. The distance from the centre of the globe at which beam of light can converge is _____ $\mathrm{mm}$.

The speed of light in media $A$ and $B$ are $2.0\times {10}^{10} \mathrm{cm} {\mathrm{s}}^{-1}$ and $1.5\times {10}^{10} \mathrm{cm} {\mathrm{s}}^{-1}$ respectively. A ray of light enters from the medium $B$ to $A$ at an incident angle $\theta$. If the ray suffers total internal reflection, then

The graph between $\frac{1}{u}$ and $\frac{1}{v}$ for a thin convex lens in order to determine its focal length is plotted as shown in the figure. The refractive index of lens is $1.5$ and its both the surfaces have same radius of curvatures $R$. The value of $R$ will be _____ $\mathrm{cm}$.

(Where $u=$ object distance, $v=$ image distance)

In the given figure, the face $AC$ of the equilateral prism is immersed in a liquid of refractive index $n$. For incident angle $60°$ at the side $AC$, the refracted light beam just grazes along face $AC$. The refractive index of the liquid $n=\frac{\sqrt{x}}{4}$. The value of $x$ is _____ .
(Given refractive index of glass $=1.5$)