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 $.....\, \text{cm}$.

An object is placed at $P$ in front of a spherical surface with refractive index $1.5$. Its image is formed at $Q$. If $PO = OQ$,find the distance $PO$. (in $R$)

$A$ curved surface of radius $R$ separates two media of refractive indices $\mu_1$ and $\mu_2$ as shown in figures $A$ and $B$. Identify the correct statement$(s)$ related to the formation of images of a real object $O$ placed at distance $x$ from the pole of the concave surface,as shown in figure $B$.

$A$ glass rod has ends as shown in the figure. The refractive index of glass is $\mu$. The object $O$ is at a distance $2R$ from the surface of larger radius of curvature. The distance between the apexes of the ends is $3R$. The range of $\mu$ for which the image is real is given by

$A$ point object is placed at a distance of $15 \, cm$ from a glass rod with a spherical end. The radius of curvature of the spherical end is $30 \, cm$. Find the position of the image formed.

$A$ point object $O$ is placed on the axis of a cylindrical piece of glass of refractive index $1.6$ as shown in the figure. One surface of the glass piece is convex with a radius of curvature $3 \,mm$. The point appears to be at $5 \,mm$ on the axis when viewed along the axis from the right side of the convex surface. The distance of the point object from the convex surface is: (in $\,mm$)