A vessel completely filled with water has two holes 'P' and 'Q' at depths ' 2 h ' and ' 8 h ' from the top…

Torricelli's Law states that the velocity of efflux from an orifice at depth $h$ is $v = \sqrt{2gh}$. The volume flow rate is given by $A v$.

For hole P at depth $2h$, the velocity is $v_P = \sqrt{2g(2h)} = 2\sqrt{gh}$. Its area is $A_P = a^2$.

For hole Q at depth $8h$, the velocity is $v_Q = \sqrt{2g(8h)} = 4\sqrt{gh}$. Its area is $A_Q = \pi r^2$.

Equating the volume flow rates:

$a^2 \cdot 2\sqrt{gh} = \pi r^2 \cdot 4\sqrt{gh}$

Dividing both sides by $2\sqrt{gh}$ gives:

$a^2 = 2\pi r^2$

Thus, $a = r\sqrt{2\pi}$, corresponding to option B.