Region $I$ and $II$ are separated by a spherical surface of radius $25\, \text{cm}$. An object is kept in region $I$ at a distance of $40\, \text{cm}$ from the surface. The distance of the image from the surface is $.....\, \text{cm}$.

$A$ spherical interface of radius $R$ separates two media of refractive indices $1$ and $1.4$ respectively, as shown in the figure below. $A$ point source is placed at a distance of $4R$ in front of the spherical interface. The magnitude of the lateral magnification of the point source image is . . . . . . .

In a long glass tube,a mixture of two liquids $A$ and $B$ with refractive indices $1.3$ and $1.4$ respectively,forms a convex refractive meniscus towards $A$. If an object placed at $13 \ \text{cm}$ from the vertex of the meniscus in $A$ forms an image with a magnification of $-2$,then the radius of curvature of the meniscus is:

$A$ point object $O$ is placed in front of a glass rod having a spherical end of radius of curvature $30 \,cm$. The image would be formed at

Light from a point source in air falls on a spherical glass surface (refractive index,$\mu=1.5$ and radius of curvature $=50\ cm$). The image is formed at a distance of $200\ cm$ from the glass surface inside the glass. The magnitude of distance of the light source from the glass surface is . . . . . . $m$.

$A$ spherical surface of radius of curvature $R$ separates air from glass of refractive index $1.5$. The centre of curvature is in the glass. $A$ point object $P$ placed in air forms a real image $Q$ in the glass. The line $PQ$ cuts the surface at point $O$ and $PO = OQ = x$. Hence the distance $x$ is equal to (in $R$)