$A$ parallel beam of light travelling in air (refractive index $1.0$) is incident on a convex spherical glass surface of radius of curvature $50 \ cm$. The refractive index of glass is $1.5$. The rays converge to a point at a distance $x \ cm$ from the centre of curvature of the spherical surface. The value of $x$ is . . . . . . $cm$.

$A$ 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.

$A$ spherical surface of radius of curvature $R$ separates air from glass (refractive index $= 1.5$). The centre of curvature is in the glass medium. $A$ point object $O$ is placed in air on the optic axis of the surface,so that its real image is formed at $I$ inside the glass. The line $OI$ intersects the spherical surface at $P$ and $PO = PI$. The distance $PO$ is equal to: (in $R$)

$A$ narrow parallel beam of light is incident paraxially on a solid transparent sphere of radius $r$ kept in air. What should be the refractive index if the beam is to be focused at the farther surface of the sphere?

Two identical glass rods $S_1$ and $S_2$ (refractive index $= 1.5$) have one convex end of radius of curvature $10 \ cm$. They are placed with the curved surfaces at a distance $d$ as shown in the figure,with their axes (shown by the dashed line) aligned. When a point source of light $P$ is placed inside rod $S_1$ on its axis at a distance of $50 \ cm$ from the curved face,the light rays emanating from it are found to be parallel to the axis inside $S_2$. The distance $d$ is (in $cm$)

$A$ spherical convex surface of power $5 \text{ dioptre}$ separates object and image space of refractive indices $1.0$ and $\frac{4}{3}$ respectively. The radius of curvature of the surface is (in $\text{ cm}$)