$A$ convex mirror of focal length $15 \ cm$ and a concave mirror of focal length $10 \ cm$ are placed facing each other at a distance of $40 \ cm$ apart. $A$ point object is placed between the mirrors on their common axis at a distance of $15 \ cm$ from the concave mirror. The position of the image formed by reflection at the convex mirror will be at a distance of ..... $cm$ from it.

$A$ 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 $n=1.48, h=85 \ cm$

$A$ bulb of $100 \text{ W}$ is hanging at a height of $1 \text{ m}$ above the centre of a circular table of diameter $4 \text{ m}$. If the intensity at a point on its rim is $I_0$,then the intensity at the centre of the table will be:

$A$ concave mirror has a radius of curvature of $40\, cm$. It is at the bottom of a glass that has water filled up to $5\, cm$ (see figure). If a small particle is floating on the surface of water,its image as seen from directly above the glass is at a distance $d$ from the surface of water. The value of $d$ is close to ......$cm$ (Refractive index of water $= 1.33$)

Which of the following statements is correct?

$A$ lamp is hanging along the axis of a circular table of radius $r$. At what height should the lamp be placed above the table,so that the illuminance at the edge of the table is $\frac{1}{8}$ of that at its center?