Planck's constant $(h),$ speed of light in vacuum $(c)$ and Newton's gravitational constant $(G)$ are three fundamental constants. Which of the following combinations of these has the dimension of length $?$

Let us consider an equation

$\frac{1}{2} m v^{2}=m g h$

where $m$ is the mass of the body. velocity, $g$ is the acceleration do gravity and $h$ is the height. whether this equation is dimensionally correct. 

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

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 -

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

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

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

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

The quantities $A$ and $B$ are related by the relation, $m = A/B$, where $m$ is the linear density and $A$ is the force. The dimensions of $B$ are of

If pressure $P$, velocity $V$ and time $T$ are taken as fundamental physical quantities, the dimensional formula of force is