The figure shows a glowing mercury tube. The illuminances at points $A$, $B$, and $C$ are related as

Shown in the figure is a convergent lens placed inside a cell filled with a liquid. The lens has a focal length of $+20\,cm$ when in air and its material has a refractive index of $1.50$. If the liquid has a refractive index of $1.60$,the focal length of the system is......$cm$.

$A$ cube of side $2\, m$ is placed in front of a concave mirror of focal length $1\, m$ with its face $P$ at a distance of $3\, m$ and face $Q$ at a distance of $5\, m$ from the mirror. The distance between the images of face $P$ and $Q$ and the heights of the images of $P$ and $Q$ are:

$A$ plano-convex lens is made of a material of refractive index $n$. When a small object is placed $30 \ cm$ away in front of the curved surface of the lens,an image of double the size of the object is produced. Due to reflection from the curved surface of the lens,another faint image is observed at a distance of $10 \ cm$ away from the lens. Which of the following statement$(s)$ is(are) true?
$(A)$ The refractive index of the lens is $2.5$
$(B)$ The radius of curvature of the convex surface is $45 \ cm$
$(C)$ The faint image is erect and real
$(D)$ The focal length of the lens is $20 \ cm$

The dispersive power of the material of a lens of focal length $20 \; cm$ is $0.08$. The longitudinal chromatic aberration of the lens is ...... $cm$.

As shown in the figure, a parallel beam of light is incident on the upper part of a prism of angle $1.8^{\circ}$ and material of refractive index $1.5$. The light emerging from the prism falls on a concave mirror of radius of curvature $40 \,cm$. The distance of the point from the principal axis of the mirror where the light rays are focused after reflection from the mirror is: