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 gap between the images formed by the two mirrors. The radius of the convex mirror is .......$cm$.

$A$ 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 $\frac{100}{21}\, cm$

$A$ 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 $^{\circ}$)

The flat bottom of a cylindrical tank is silvered and water $(\mu = 4/3)$ is filled in the tank up to a height $h$. $A$ small bird is hovering at a height $3h$ from the bottom of the tank. When a small hole is opened near the bottom of the tank,the water level falls at the rate of $1\, cm/s$. The bird will perceive that its image's velocity is:

$A$ concave mirror of radius of curvature $h$ is placed at the bottom of a tank containing a liquid of refractive index $\mu$ up to a depth $d$. An object $P$ is placed at a height $h$ above the bottom of the mirror. Outside the liquid,an observer $O$ views the object and its image in the mirror. The apparent distance between these two will be

Four bulbs,red,green,white and blue (denoted by $R, G, W$ and $B$ respectively) are kept in front of a converging lens (as shown in the figure). The observer sees that the green and blue bulbs are kept to the left of the principal axis,while the red and white bulbs are kept to the right of the principal axis. He also sees that the red and green bulbs are above the principal axis,while the white and blue bulbs are below the principal axis. The screens $S_1$ and $S_2$ are set at appropriate positions for focusing to view the images. Choose the figure that correctly represents the images as seen by the observer.