$A$ light pointer fixed to one prong of a tuning fork touches a vertical plate. The fork is set vibrating and the plate is allowed to fall freely. If eight oscillations are counted when the plate falls through $10 \, cm$,the frequency of the tuning fork is .... $Hz$

An observer is standing $500 ~m$ away from a vertical hill. Starting between the observer and the hill,a police van sounding a siren of frequency $1000 ~Hz$ moves towards the hill with a uniform speed. If the frequency of the sound heard directly from the siren is $970 ~Hz$,the frequency of the sound heard after reflection from the hill (in $Hz$) is about,(velocity of sound $= 330 ~m/s$):

$A$ function $y = f(x)$ is given by $x = \frac{1}{1 + t^2}$ and $y = \frac{1}{t(1 + t^2)}$ for all $t > 0$. Then $f$ is:

The number of oxygen atoms bonded to one phosphorous atom in $P_4O_6$ is

If $p$-th,$q$-th,and $r$-th terms of a geometric progression are the positive numbers $a, b,$ and $c$ respectively,then the angle between the vectors $(\log a^2) i + (\log b^2) j + (\log c^2) k$ and $(q-r) i + (r-p) j + (p-q) k$ is

Two thin lenses of focal lengths $f_1$ and $f_2$ are in contact and coaxial. The power of the combination is