$A$ hoop of radius $r$ and mass $m$ rotating with an angular velocity $\omega_0$ is placed on a rough horizontal surface. The initial velocity of the centre of the hoop is zero. What will be the velocity of the centre of the hoop when it ceases to slip?

Choose the incorrect statement from the following:
$S1:$ The efficiency of a heat engine can be $1$,but the coefficient of performance of a refrigerator can never be infinity.
$S2:$ The first law of thermodynamics is basically the principle of conservation of energy.
$S3:$ The second law of thermodynamics does not allow several phenomena consistent with the first law.
$S4:$ $A$ process,whose only result is to transfer heat from a colder to a hotter object,is impossible.

In the figure, a potential of $+1200 \, V$ is applied to point $A$ and point $B$ is earthed. What is the potential at point $P$ (in $V$)?

What will be the $pH$ of an aqueous solution of $1.0 \ M$ ammonium formate? (Given: $pK_{a}=3.8$ and $pK_{b}=4.8$)

$A$ steel meter scale is to be ruled so that millimeter intervals are accurate within about $5 \times 10^{-5} \ m$ at a certain temperature. The maximum temperature variation allowable during the ruling is (Coefficient of linear expansion of steel $= 10 \times 10^{-6} \ K^{-1}$) (in $^{\circ} C$)

An uncharged capacitor is connected to a battery. On charging the capacitor,