$A$ parallel beam of light travelling in air (refractive index $1.0$) is incident on a convex spherical glass surface of radius of curvature $50 \ cm$. The refractive index of glass is $1.5$. The rays converge to a point at a distance $x \ cm$ from the centre of curvature of the spherical surface. The value of $x$ is . . . . . . $cm$.

Two concave refracting surfaces of equal radii of curvature $R$ and refractive index $1.5$ face each other in air as shown in the figure. $A$ point object $O$ is placed midway between $P$ and $B$. The separation between the images of $O$ formed by each refracting surface is: (in $R$)

Parallel rays are incident on a transparent sphere along its one diameter. After refraction,these rays converge at the other end of this diameter. The refractive index of the sphere is:

$A$ hemispherical glass lens with a refractive index of $1.5$ is placed in a liquid with a refractive index of $1.3$ (see the figure). The radius of the hemispherical lens is $10 \,cm$. $A$ parallel beam of light traveling in the liquid is refracted by the glass lens. The absolute value of the position of the image from the center of the glass lens will be (in $\,cm$)

$A$ point object is placed at the center of a glass sphere of radius $6 \ cm$ and refractive index $1.5$. The distance of the virtual image from the surface of the sphere is.......$cm$

The image of an object placed in air formed by a convex refracting surface is at a distance of $10 \ m$ behind the surface. The image is real and is at $\frac{2}{3}$ of the distance of the object from the surface. The wavelength of light inside the surface is $\frac{2}{3}$ times the wavelength in air. The radius of the curved surface is $\frac{x}{13} \ m$. The value of $x$ is ....... .