$A$ thin spherical conducting shell of radius $R$ has a charge $q$. Another charge $Q$ is placed at the centre of the shell. The electrostatic potential at a point $P$ at a distance $R/2$ from the centre of the shell is

Power delivered to a mass is $P = (3t^2 - 2t + 1) \text{ W}$. The change in kinetic energy from $t = 2 \text{ s}$ to $t = 4 \text{ s}$ is $...... \text{ J}$.

The number of unit cells present in $39 \ g$ of potassium if it crystallizes as body-centered cubic $(BCC)$ is ($N = \text{Avogadro number}$,$\text{At. wt. of potassium} = 39$).

$A$ conducting rod $AC$ of length $4l$ is rotated about a point $O$ in a uniform magnetic field $\vec B$ directed into the paper. $AO = l$ and $OC = 3l$. Then

The air density at Mount Everest is less than that at sea level. Mountaineers find that for one trip lasting a few hours,the extra oxygen needed corresponds to $30,000 \ cc$ at sea level (pressure $1 \ atm$,temperature $27^{\circ}C$). Assuming that the temperature around Mount Everest is $-73^{\circ}C$ and that the oxygen cylinder has a capacity of $5.2 \ L$,the pressure at which $O_2$ must be filled in the cylinder is ..... $atm$.

The oxoanion which contains all equivalent $M-O$ bonds is:
$(I)$ $CrO_4^{2-}$
$(II)$ $MnO_4^{-}$
$(III)$ $Cr_2O_7^{2-}$