Consider a point object $O$ situated at a distance of $30 \ cm$ from the centre of a sphere of radius $2 \ cm$ and refractive index $\mu_2 = 1.5$ as shown in the figure. If the refractive index of the region surrounding this sphere is $\mu_1 = 1.4$, then the position of the image due to refraction by the sphere with respect to the centre is:

$A$ laser beam falls on a crystal ball of radius $R$ as shown in the figure. What is its refractive index?

$A$ parallel beam of light is allowed to fall on a transparent spherical globe of diameter $30 \,cm$ and refractive index $1.5$. The distance from the centre of the globe at which the beam of light can converge is . . . . . . $mm$.

$A$ denser medium of refractive index $1.5$ has a concave surface of radius of curvature $12 \, cm$. An object is situated in the denser medium at a distance of $9 \, cm$ from the pole. Locate the image due to refraction in air.

The figure shows a transparent sphere of radius $R$ and refractive index $\mu$. An object $O$ is placed at a distance $x$ from the pole of the first surface so that a real image is formed at the pole of the exactly opposite surface. If an object is placed at a distance $R$ from the pole of the first surface,then the real image is formed at a distance $R$ from the pole of the second surface. The refractive index $\mu$ of the sphere is given by

Three glass cylinders of equal height $H = 30 \text{ cm}$ and same refractive index $n = 1.5$ are placed on a horizontal surface as shown in the figure. Cylinder $I$ has a flat top,cylinder $II$ has a convex top,and cylinder $III$ has a concave top. The radii of curvature of the two curved tops are same $(R = 3 \text{ m})$. If $H_1, H_2$ and $H_3$ are the apparent depths of a point $X$ on the bottom of the three cylinders,respectively,the correct statement$(s)$ is/are:
$(1) H_3 > H_1$
$(2) 0.8 \text{ cm} < (H_2 - H_1) < 0.9 \text{ cm}$
$(3) H_2 > H_3$
$(4) H_2 > H_1$