$A, B, C$ and $D$ are four different physical quantities having different dimensions. None of them is dimensionless. We know that the equation $AD = C \ln(BD)$ holds true. Which of the following combinations is not a meaningful quantity?

If energy $(E)$,velocity $(v)$,and force $(F)$ are taken as fundamental quantities,what are the dimensions of mass?

If the equation $y = x^2 r + M^1 L^1 T^{-2}$ is dimensionally correct,find the dimensional formula of $x^2$. (Here,$r$ represents displacement.)

$A$ force is represented by $F = ax^2 + bt^{1/2}$,where $x$ is distance and $t$ is time. The dimensions of $b^2/a$ are:

$A$ small steel ball of radius $r$ is allowed to fall under gravity through a column of a viscous liquid of coefficient of viscosity $\eta$. After some time,the velocity of the ball attains a constant value known as terminal velocity $v_T$. The terminal velocity depends on $(i)$ the mass of the ball $m$,$(ii)$ $\eta$,$(iii)$ $r$,and $(iv)$ acceleration due to gravity $g$. Which of the following relations is dimensionally correct?

The efficiency of an engine is given by $\eta = \frac{\alpha \beta}{\sin \theta} \cdot \log_{e} \frac{\beta x}{kT}$,where $\alpha$ and $\beta$ are constants. If $T$ is the absolute temperature,$k$ is the Boltzmann constant,$\theta$ is angular displacement,and $x$ is distance,then the incorrect statement is: