A force defined by $F=\alpha t^2+\beta t$ acts on a particle at a given time $t$. The factor which is dimensionless, if $\alpha$ and $\beta$ are constants, is:

The dimensions of power are

The quantities $\quad x=\frac{1}{\sqrt{\mu_{0} \epsilon_{0}}}, y=\frac{E}{B}$ and $z=\frac{l}{C R}$ are defined where $C-$ capacitance $R-$Resistance, $l-$length, $E-$Electric field, $B-$magnetic field and $\varepsilon_{0}, \mu_{0},$ -free space permittivity and permeability respectively. Then....

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

If $L$ and $R$ are respectively the inductance and resistance, then the dimensions of $\frac{L}{R}$ will be

The dimensions of Stefan-Boltzmann's constant $\sigma$ can be written in terms of Planck's constant $h$, Boltzmann's constant $k_B$ and the speed of light $c$ as $\sigma=h^\alpha k_B^\beta c^\gamma$. Here,