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 $R$)

$A$ transparent thin film of uniform thickness and refractive index $n_1=1.4$ is coated on the convex spherical surface of radius $R$ at one end of a long solid glass cylinder of refractive index $n_2=1.5$,as shown in the figure. Rays of light parallel to the axis of the cylinder traversing through the film from air to glass get focused at distance $f_1$ from the film,while rays of light traversing from glass to air get focused at distance $f_2$ from the film. Then:
$(A)$ $|f_1|=3R$
$(B)$ $|f_1|=2.8R$
$(C)$ $|f_2|=2R$
$(D)$ $|f_2|=1.4R$

$A$ beam of diameter $d$ is incident on a glass hemisphere as shown. If the radius of curvature of the hemisphere is very large in comparison to $d$,then the diameter of the beam at the base of the hemisphere will be:

Light from a point source in air falls on a spherical glass surface (refractive index,$\mu=1.5$ and radius of curvature $=50\ cm$). The image is formed at a distance of $200\ cm$ from the glass surface inside the glass. The magnitude of distance of the light source from the glass surface is . . . . . . $m$.

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:

$A$ concave spherical refracting surface separates two media,glass and air $(\mu_1 = 1.5, \mu_2 = 1.0)$. If the image is to be real,at what minimum distance $u$ should the object be placed in the glass if $R$ is the radius of curvature?