$A$ lens with a focal length of $50 \ cm$ forms an image of a distant object that subtends an angle of $1 \text{ milliradian}$ at the lens. What is the size of the image in $mm$?

$A$ lens placed $10 \,cm$ away from a wall casts a sharp inverted image of a candle on it. It again casts a sharp image when the lens is moved $20 \,cm$ further away from the wall. Now,the candle and the lens are moved such that a sharp inverted image with unit magnification is formed on the wall. To achieve this configuration,the candle was moved

$A$ bi-concave symmetric lens made of glass has a refractive index of $1.5$. It has both surfaces with the same radius of curvature $R$. On immersion in a liquid of refractive index $1.75$,it will behave as a:

$A$ thin lens of refractive index $1.5$ has an optical power of $-5 \ D$ in air. Its optical power in a liquid medium with a refractive index of $1.6$ is $D$.

$A$ convex lens has power $P$. It is cut into two halves along its principal axis. Further,one piece (out of the two halves) is cut into two halves perpendicular to the principal axis (as shown in the figure). Choose the incorrect option for the reported pieces.

Images of same size are formed by a convex lens when an object is placed either at $20 \ cm$ or $10 \ cm$ distance from the lens. The focal length of the lens is (in $cm$)