$A$ particle is oscillating on the $X-$ axis with an amplitude $2\, cm$ about the point $x_0 = 10\, cm$ with a frequency $\omega $. $A$ concave mirror of focal length $5\, cm$ is placed at the origin (see figure). Identify the correct statements.
$(A)$ The image executes periodic motion
$(B)$ The image executes non-periodic motion
$(C)$ The turning points of the image are asymmetric w.r.t the image of the point at $x = 10\, cm$
$(D)$ The distance between the turning points of the oscillation of the image is $\frac{100}{21}\, cm$

An electric lamp is fixed at the ceiling of a circular tunnel as shown in the figure. What is the ratio of the intensities of light at base $A$ and a point $B$ on the wall?

An object is placed $60 \ cm$ in front of a convex mirror of focal length $30 \ cm$. $A$ plane mirror is now placed facing the object in between the object and the convex mirror such that it covers the lower half of the convex mirror. What should be the distance of the plane mirror from the object so that there will be no parallax between the images formed by the two mirrors (in $cm$)?

$A$ solid hemisphere of refractive index $\mu_1 = 3/2$ and radius $R = 20\ cm$ is placed as shown in the figure. Another special cylindrical glass of refractive index $\mu_2$,radius $R$,and thickness $t = 2R/3$ is placed adjacent to the flat surface of the solid hemisphere. There is a very small luminous spot at $P$ just inside the solid hemisphere,and the $MN$ surface of the special glass is silvered. Considering only paraxial rays,if the final image of point $P$ for an observer to the left of the hemisphere is at the center $C$ of the hemisphere,find the refractive index $\mu_2$ of the special glass.

An object is placed in front of a convex mirror at a distance of $50\, cm$. $A$ plane mirror is introduced covering the lower half of the convex mirror. If the distance between the object and the plane mirror is $30\, cm$,it is found that there is no parallax between the images formed by the two mirrors. The radius of curvature of the convex mirror will be......$cm$.

$A$ point source of light is $60\, cm$ from a screen and is kept at the focus of a concave mirror which reflects light on the screen. The focal length of the mirror is $20\, cm$. The ratio of average intensities of the illumination on the screen when the mirror is present and when the mirror is removed is: (in $: 1$)