$A$ system has basic dimensions as density $[D]$,velocity $[V]$,and area $[A]$. The dimensional representation of force in this system is:

$A$ calorie is a unit of heat or energy and it equals about $4.2 \; J$ where $1 \; J = 1 \; kg \; m^2 \; s^{-2}$. Suppose we employ a system of units in which the unit of mass equals $\alpha \; kg$, the unit of length equals $\beta \; m$, and the unit of time equals $\gamma \; s$. Show that a calorie has a magnitude $4.2 \; \alpha^{-1} \beta^{-2} \gamma^2$ in terms of the new units.

If $\varepsilon_0$ is the permittivity of free space,$e$ is the charge of a proton,$G$ is the universal gravitational constant,and $m_p$ is the mass of a proton,then the dimensional formula for $\frac{e^2}{4 \pi \varepsilon_0 G m_p^2}$ is:

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:

In a practical unit system,if the unit of mass becomes double and that of time becomes half,then $8 \text{ joule}$ will be equal to .............. unit of work.

The potential energy of a particle varies with distance $x$ from a fixed origin as $U = \frac{A\sqrt{x}}{x^2 + B}$,where $A$ and $B$ are dimensional constants. Find the dimensional formula for $A/B$.