$A$ 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?

An optical arrangement consists of two concave mirrors $M_1$ and $M_2$,and a convex lens $L$ with a common principal axis,as shown in the figure. The focal length of $L$ is $10 \text{ cm}$. The radii of curvature of $M_1$ and $M_2$ are $20 \text{ cm}$ and $24 \text{ cm}$,respectively. The distance between $L$ and $M_2$ is $20 \text{ cm}$. $A$ point object $S$ is placed at the mid-point between $L$ and $M_2$ on the axis. When the distance between $L$ and $M_1$ is $n/7 \text{ cm}$,one of the images coincides with $S$. The value of $n$ is. . . .

An object $O$ is placed in front of a small plane mirror $M_1$ and a large convex mirror $M_2$ of focal length $f$. The distance between $O$ and $M_1$ is $x$,and the distance between $M_1$ and $M_2$ is $y$. The images of $O$ formed by $M_1$ and $M_2$ coincide. The magnitude of $f$ is

In the diagram shown,all the velocities are given with respect to the earth. What is the relative velocity of the image in mirror $(1)$ with respect to the image in mirror $(2)$? The mirror $(1)$ forms an angle $\beta$ with the vertical.

$A$ plane mirror is placed $22.5\; cm$ in front of a concave mirror of focal length $10\; cm$. An object is placed between the two mirrors such that the images formed by both mirrors coincide. Find the magnification of the image produced by the concave mirror.

$A$ glass beaker has a solid,plano-convex base of refractive index $1.60$,as shown in the figure. The radius of curvature of the convex surface $(SPU)$ is $9 \ cm$,while the planar surface $(STU)$ acts as a mirror. This beaker is filled with a liquid of refractive index $n$ up to the level $QPR$. If the image of a point object $O$ at a height of $h$ ($OT$ in the figure) is formed onto itself,then which of the following option$(s)$ is (are) correct?
$(A)$ For $n=1.42, h=50 \ cm$
$(B)$ For $n=1.35, h=36 \ cm$
$(C)$ For $n=1.45, h=65 \ cm$
$(D)$ For $n=1.48, h=85 \ cm$