$A$ 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?

$A$ point source of $100$ candela is held $5\,m$ above a sheet of blotting paper which reflects $75\%$ of light incident upon it. The illuminance of the blotting paper is

In the figure shown,if the object $O$ moves towards the plane mirror $M_1$,then the image $I$ (which is formed after successive reflections from $M_1$ and $M_2$ respectively) will move:

$A$ 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.

$A$ lamp rated at $100 \, cd$ hangs over the middle of a round table with diameter $3 \, m$ at a height of $2 \, m$. It is replaced by a lamp of $25 \, cd$ and the distance to the table is changed so that the illumination at the centre of the table remains as before. The illumination at the edge of the table becomes $X$ times the original. Then $X$ is

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