$A$ hemispherical glass body of radius $10\, cm$ and refractive index $1.5$ is silvered on its curved surface. $A$ small air bubble is $6\, cm$ below the flat surface inside it along the axis. The position of the image of the air bubble made by the mirror is seen:

$A$ horizontal parallel beam of light passes through a vertical convex lens of focal length $20 \,cm$ and is then reflected by a tilted plane mirror,so that it converges to a point $I$. The distance $P I$ is $10 \,cm$. $M$ is a point at which the axis of the lens intersects the mirror. The distance $P M$ is $10 \,cm$. The angle which the mirror makes with the horizontal is (in $^{\circ}$)

$A$ point object is kept at the first focus of a convex lens. If the lens starts moving towards the right with a constant velocity, the image will

$A$ point source is placed at a distance of $15 \, cm$ from a converging (convex) lens of focal length $10 \, cm$. At what distance (in $cm$) should a convex mirror of focal length $12 \, cm$ be placed so that the image is formed on the object itself?

$A$ beaker of radius $r$ is filled with water (refractive index $\frac{4}{3}$) up to a height $H$ as shown in the figure. The beaker is kept on a horizontal table rotating with angular speed $\omega$. This makes the water surface curved so that the difference in the height of the water level at the center and at the circumference of the beaker is $h$ $(h \ll H, h \ll r)$,as shown in the figure. Take this surface to be approximately spherical with a radius of curvature $R$. Which of the following is/are correct? ($g$ is the acceleration due to gravity)
$(A)$ $R=\frac{h^2+r^2}{2 h}$
$(B)$ $R=\frac{r^2}{2 h}$
$(C)$ Apparent depth of the bottom of the beaker is close to $\frac{3 H}{4}\left(1+\frac{\omega^2 H}{4 g}\right)^{-1}$
$(D)$ Apparent depth of the bottom of the beaker is close to $\frac{3 H}{2}\left(1+\frac{\omega^2 H}{2 g}\right)^{-1}$

The object distance $u$,the image distance $v$,and the magnification $m$ in a lens follow certain linear relations. These are: