$A$ particle of mass $m$ is moving in a circular path of constant radius $r$ such that its centripetal acceleration $a_c$ is varying with time $t$ as $a_c = k^2rt^2$. The power delivered to the particle by the forces acting on it is

The shortest distance between the line $y - x = 1$ and the curve $x = y^2$ is:

$A$ man is watching two trains,one leaving and the other coming with equal speed of $4\,m/s$. If they sound their whistles each of frequency $240\,Hz$,the number of beats per second heard by the man will be equal to (velocity of sound in air $= 320\,m/s$)

Two ideal slits $S_1$ and $S_2$ are at a distance $d$ apart and are illuminated by light of wavelength $\lambda$ passing through an ideal source slit $S$ placed on the line through $S_2$ as shown. The distance between the plane of the slits and the source slit is $D$. $A$ screen is held at a distance $D$ from the plane of the slits. The minimum value of $d$ for which there is darkness at $O$ is

The limiting friction between two bodies in contact is independent of

Ethanol,when reacted with $PCl_5$,gives $A$,$POCl_3$,and $HCl$. $A$ reacts with silver nitrite to form $B$ (major product) and $AgCl$. $A$ and $B$ are respectively: