$A$ clear transparent glass sphere $(\mu=1.5)$ of radius $R$ is immersed in a liquid of refractive index $1.25$. $A$ parallel beam of light incident on it will converge to a point. The distance of this point from the center will be

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:

The slab of a material of refractive index $2$ shown in the figure has a curved surface $APB$ of radius of curvature $10 \, cm$ and a plane surface $CD$. On the left of $APB$ is air and on the right of $CD$ is water with refractive indices as given in the figure. An object $O$ is placed at a distance of $15 \, cm$ from the pole $P$ as shown. The distance of the final image of $O$ from $P$,as viewed from the left,is ..... $cm$.

As shown in the figure,an air bubble is located at a distance of $3 \, cm$ from a spherical surface of a glass $(\mu = 1.5)$ with a diameter of $10 \, cm$. If the surface is concave,at what distance from the surface will the bubble appear? (in $cm$)

$A$ quarter cylinder of radius $R$ and refractive index $1.5$ is placed on a table. $A$ point object $P$ is kept at a distance of $mR$ from it. Find the value of $m$ for which a ray from $P$ will emerge parallel to the table as shown in the figure.

$A$ curved surface of radius $R$ separates two media of refractive indices $\mu_1$ and $\mu_2$ as shown in figures $A$ and $B$. Choose the correct statement$(s)$ related to the virtual image formed by object $O$ placed at a distance $x$,as shown in figure $A$.