Three glass cylinders of equal height $\mathrm{H}=30\mathrm{ }\mathrm{c}\mathrm{m}$ and same refractive index $\mathrm{n}=1.5$ are placed on a horizontal surface shown in figure. Cylinder $\mathrm{I}$ has a flat top, cylinder $\mathrm{I}\mathrm{I}$ has a convex top and cylinder $\mathrm{I}\mathrm{I}\mathrm{I}$ has a concave top. The radii of curvature of the two curved tops are same $\left(\mathrm{R}=3 \mathrm{m}\right).$ If ${\mathrm{H}}_1,{\mathrm{H}}_2$ and ${\mathrm{H}}_3$ are the apparent depths of a point $\mathrm{X}$ on the bottom of the three cylinders, respectively, the correct statement(s) is/are

A small object is placed $50 \mathrm{cm}$ to the left of thin convex lens of focal length $30 \mathrm{cm}$. A convex spherical mirror of radius of curvature $100 \mathrm{cm}$ is placed to the right of the lens at a distance of $50 \mathrm{cm}$. The mirror is tilted such that the axis of the mirror is at an angle $\theta =30°$ to the axis of the lens, as shown in the figure. If the origin of the coordinate system is taken to be at the centre of the lens, the coordinates (in $\mathrm{cm}$) of the point $(x, y)$ at which the image is formed are:

Light guidance in an optical fiber can be understood by considering a structure comprising of thin solid glass cylinder of refractive index $n_1$ surrounded by a medium of lower refractive index $n_2$. The light guidance in the structure takes place due to successive total internal reflections at the interface of the media $n_1$ and $n_2$ as shown in the figure. All rays with the angle of incidence $i$ less than a particular value $i_m$ are confined in the medium of refractive index $n_1$ . The numerical aperture (NA) of the structure is defined as $\sin i_m$.

For two structures namely $S_1$ with $n_1=\frac{\sqrt{45}}{4}$ and $n_2=\frac{3}{2}$ , and $S_2$ with $n_1=\frac{8}{5}$ and $n_2=\frac{7}{5}$ and taking the refractive index of water to be $\frac{4}{3}$ and that of air to be 1, the correct option(s) is (are)

Light guidance in an optical fiber can be understood by considering a structure comprising of thin solid glass cylinder of refractive index $n_1$ surrounded by a medium of lower refractive index $n_2$. The light guidance in the structure takes place due to successive total internal reflections at the interface of the media $n_1$ and $n_2$ as shown in the figure. All rays with the angle of incidence $i$ less than a particular value $i_m$ are confined in the medium of refractive index $n_1$ . The numerical aperture (NA) of the structure is defined as $\sin i_m$.

If two structures of same cross-sectional area, but different numerical apertures $\mathrm{N}{\mathrm{A}}_1$ and $\mathrm{N}{\mathrm{A}}_2$ $\left(\mathrm{N}{\mathrm{A}}_2<\mathrm{N}{\mathrm{A}}_1\right)$ are joined longitudinally, the numerical aperture of the combined structure is

An extended object is placed at point $O$, $10 \mathrm{cm}$ in front of a convex lens $L_1$ and a concave lens $L_2$ is placed $10 \mathrm{cm}$ behind it, as shown in the figure. The radii of curvature of all the curved surfaces in both the lenses are $20 \mathrm{cm}$. The refractive index of both the lenses is $1.5$. The total magnification of this lens system is: