Radii of curvature of a converging lens are in the ratio $1 : 2$. Its focal length is $6 \,cm$ and refractive index is $1.5$. Then its radii of curvature are ......... respectively.

$A$ plano-convex thin lens is used to obtain the image of a point object $O$ on the screen $S$ as shown in the figure. The thickness of the lens in the middle is $0.5 \ cm$ and the refractive index of the material of the lens is $1.5$. If the separation $D$ between the object and the screen is to be minimum,then the aperture diameter of the lens should be.....$cm$.

$A$ 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

An object is placed at a point distant $x$ from the focus of a convex lens and its image is formed at $I$ as shown in the figure. The distances $x$ and $x'$ satisfy the relation:

$A$ convex lens is made of $3$ layers of glass of $3$ different materials as shown in the figure. $A$ point object is placed on its axis. The number of images of the object formed is:

The magnifications produced by a convex lens for two positions of an object are $4$ and $3$, respectively. If the distance of separation between the two positions of the object is $2 \,cm$, then the focal length of the lens is (in $\,cm$)