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:

Force $(F)$ and density $(d)$ are related as $F\, = \,\frac{\alpha }{{\beta \, + \,\sqrt d }}$ then dimension of $\alpha $ are

To find the distance $d$ over which a signal can be seen clearly in foggy conditions, a railways engineer uses dimensional analysis and assumes that the distance depends on the mass density $\rho$ of the fog, intensity (power/area) $S$ of the light from the signal and its frequency $f$. The engineer find that $d$ is proportional to $S ^{1 / n}$. The value of $n$ is:

Stokes' law states that the viscous drag force $F$ experienced by a sphere of radius $a$, moving with a speed $v$ through a fluid with coefficient of viscosity $\eta$, is given by $F=6 \pi \eta a v$.If this fluid is flowing through a cylindrical pipe of radius $r$, length $l$ and a pressure difference of $p$ across its two ends, then the volume of water $V$ which flows through the pipe in time $t$ can be written as

$\frac{v}{t}=k\left(\frac{p}{l}\right)^a \eta^b r^c$

where, $k$ is a dimensionless constant. Correct value of $a, b$ and $c$ are

If $P$ represents radiation pressure, $c$ represents speed of light and $Q$ represents radiation energy striking a unit area per second, then non-zero integers $x,\,y$ and $z$ such that ${P^x}{Q^y}{c^z}$ is dimensionless, are