The dimensional formula for $r.m.s.$ (root mean square) velocity is

A book with many printing errors contains four different formulas for the displacement $y$ of a particle undergoing a certain periodic motion:

$(a)\;y=a \sin \left(\frac{2 \pi t}{T}\right)$

$(b)\;y=a \sin v t$

$(c)\;y=\left(\frac{a}{T}\right) \sin \frac{t}{a}$

$(d)\;y=(a \sqrt{2})\left(\sin \frac{2 \pi t}{T}+\cos \frac{2 \pi t}{T}\right)$

$(a=$ maximum displacement of the particle, $v=$ speed of the particle. $T=$ time-period of motion). Rule out the wrong formulas on dimensional grounds.

Identify the pair of physical quantities which have different dimensions 

The dimensional formula for Boltzmann's constant is

The equation of state of some gases can be expressed as $\left( {P + \frac{a}{{{V^2}}}} \right) = \frac{{b\theta }}{l}$ Where $P$ is the pressure, $V$ the volume, $\theta $ the absolute temperature and $a$ and $b$ are constants. The dimensional formula of $a$ is

The dimensions $\left[ MLT ^{-2} A ^{-2}\right]$ belong to the :