If ${E}, {L}, {m}$ and ${G}$ denote the quantities as energy, angular momentum, mass and constant of gravitation respectively, then the dimensions of ${P}$ in the formula ${P}={EL}^{2} {m}^{-5} {G}^{-2}$ are

From the equation $\tan \theta = \frac{{rg}}{{{v^2}}}$, one can obtain the angle of banking $\theta $ for a cyclist taking a curve (the symbols have their usual meanings). Then say, it is

In the equation $y = pq$ $tan\,(qt)$, $y$ represents position, $p$ and $q$ are unknown physical quantities and $t$ is time. Dimensional formula of $p$ is

Which of the following relation cannot be deduced using dimensional analysis? [the symbols have their usual meanings]

According to Newton, the viscous force acting between liquid layers of area $A$ and velocity gradient $\Delta v/\Delta z$ is given by $F = - \eta A\frac{{\Delta v}}{{\Delta z}}$ where $\eta $ is constant called coefficient of viscosity. The dimension of $\eta $ are

Sometimes it is convenient to construct a system of units so that all quantities can be expressed in terms of only one physical quantity. In one such system, dimensions of different quantities are given in terms of a quantity $X$ as follows: [position $]=\left[X^\alpha\right] ;[$ speed $]=\left[X^\beta\right]$; [acceleration $]=\left[X^{ p }\right]$; [linear momentum $]=\left[X^{ q }\right]$; [force $]=\left[X^{ I }\right]$. Then -

$(A)$ $\alpha+p=2 \beta$

$(B)$ $p+q-r=\beta$

$(C)$ $p-q+r=\alpha$

$(D)$ $p+q+r=\beta$