The sun subtends an angle of $0.5^{\circ}$ at the earth's surface. Its image is formed by a convex lens of focal length $50 \ \text{cm}$. The diameter of the image will be (in $\text{mm}$)

$A$ double convex lens has power $P$ and same radii of curvature $R$ of both the surfaces. The radius of curvature of a surface of a plano-convex lens made of the same material with power $1.5P$ is ......$R$.

$A$ screen is placed $90\, cm$ from an object. The image of the object on the screen is formed by a convex lens at two different locations separated by $20\, cm$. The focal length of the lens is.....$ cm$

The focal length of a plano-convex lens is equal to its radius of curvature. Find the refractive index of its material.

The distance between an object and its $3$ times magnified virtual image produced by a convex lens is $20 \,cm$. The focal length of the lens used is: (in $\,cm$)

$A$ lens is placed between a source of light and a wall. It forms images of area $A_1$ and $A_2$ on the wall for its two different positions. The area of the source of light is