$A$ gas bubble formed from an explosion under water oscillates with a period $T$,proportional to $P^{a} d^{b} E^{c}$,where $P$ is the static pressure,$d$ is the density of water,and $E$ is the energy of explosion. Then $a, b, c$ are,respectively:

The characteristic distance at which quantum gravitational effects are significant,the Planck length,can be determined from a suitable combination of the fundamental physical constants $G, h$ and $c$. Which of the following correctly gives the Planck length?

The $SI$ unit of energy is $J = kg \, m^{2} \, s^{-2}$; that of speed $v$ is $m \, s^{-1}$ and of acceleration $a$ is $m \, s^{-2}$. Which of the formulae for kinetic energy $(K)$ given below can you rule out on the basis of dimensional arguments ($m$ stands for the mass of the body):
$(a)$ $K = m^{2} v^{3}$
$(b)$ $K = (1/2) m v^{2}$
$(c)$ $K = m a$
$(d)$ $K = (3/16) m v^{2}$
$(e)$ $K = (1/2) m v^{2} + m a$

The equation of a wave is given by $Y = A \sin \omega \left( \frac{x}{v} - k \right)$,where $\omega$ is the angular velocity and $v$ is the linear velocity. The dimension of $k$ is

The dimensions of physical quantity $X$ in the equation $\text{Force} = \frac{X}{\text{Density}}$ are given by:

Which of the following is a dimensional constant?