$A$ converging beam of rays is incident on a diverging lens. Having passed through the lens,the rays intersect at a point $15\, cm$ from the lens on the opposite side. If the lens is removed,the point where the rays meet will move $5\, cm$ closer to the lens. The focal length of the lens is $......\, cm$.

$A$ thin converging lens of focal length $f=25 \,cm$ forms the image of an object on a screen placed at a distance of $75 \,cm$ from the lens. The screen is moved closer to the lens by a distance of $25 \,cm$. The distance through which the object has to be shifted, so that its image on the screen is sharp again, is (in $\,cm$)

What is the refractive index of the material of a double convex lens having radii of curvature of $5 \ cm$ and $10 \ cm$ and focal length of $\frac{20}{3} \ cm$?

The image of a small electric bulb fixed on the wall of a room is to be obtained on the opposite wall $3\, m$ away by means of a large convex lens. What is the maximum possible focal length of the lens required for the purpose?

$A$ convex lens of refractive index $\frac{3}{2}$ has a power of $2.5 \ D$. If it is placed in a liquid of refractive index $2$, the new power of the lens is: (in $D$)

$A$ 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.....$cm$.