$A$ concave mirror and a converging lens (glass with $\mu = 1.5$) both have a focal length of $3 \, cm$ when in air. When they are in water $\left( \mu = \frac{4}{3} \right)$,what are their new focal lengths?

$A$ right-angled prism of refractive index $\mu_1$ is placed in a rectangular block of refractive index $\mu_2$,which is surrounded by a medium of refractive index $\mu_3$,as shown in the figure. $A$ ray of light 'e' enters the rectangular block at normal incidence. Depending upon the relationships between $\mu_1, \mu_2$ and $\mu_3$,it takes one of the four possible paths '$ef$','$eg$','$eh$' or '$ei$'. Match the paths in List-$I$ with conditions of refractive indices in List-$II$ and select the correct answer using the codes given below the lists:

List-$I$	List-$II$
$P$. $e \rightarrow f$	$1$. $\mu_1 > \sqrt{2} \mu_2$
$Q$. $e \rightarrow g$	$2$. $\mu_2 > \mu_1$ and $\mu_2 > \mu_3$
$R$. $e \rightarrow h$	$3$. $\mu_1 = \mu_2$
$S$. $e \rightarrow i$	$4$. $\mu_2 < \mu_1 < \sqrt{2} \mu_2$ and $\mu_2 > \mu_3$

Codes: $P \quad Q \quad R \quad S$

When a pin is moved along the principal axis of a small concave mirror,the image position coincides with the object at a point $0.5\, m$ from the mirror. If the mirror is placed at a depth of $0.2\, m$ in a transparent liquid,the same phenomenon occurs when the pin is placed $0.4\, m$ from the mirror. The refractive index of the liquid is

Write the names of instruments developed by the use of properties of reflection and refraction of mirrors,lenses,and prisms.

$A$ cube of side $2\, m$ is placed in front of a concave mirror of focal length $1\, m$ with its face $P$ at a distance of $3\, m$ and face $Q$ at a distance of $5\, m$ from the mirror. The distance between the images of face $P$ and $Q$ and the heights of the images of $P$ and $Q$ are:

$A$ 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 $\frac{3 H}{2}\left(1+\frac{\omega^2 H}{2 g}\right)^{-1}$