The electrical resistance $R$ of a conductor of length $l$ and area of cross section $a$ is given by $R = \frac{{\rho l}}{a}$ where $\rho$ is the electrical resistivity. What is the dimensional formula for electrical conductivity $\sigma $ which is reciprocal of resistivity?

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

If force $[F],$ acceleration $[A]$ and time $[T]$ are chosen as the fundamental physical quantities. Find the dimensions of energy.

A massive black hole of mass $m$ and radius $R$ is spinning with angular velocity $\omega$. The power $P$ radiated by it as gravitational waves is given by $P=G c^{-5} m^{x} R^{y} \omega^{z}$, where $c$ and $G$ are speed of light in free space and the universal gravitational constant, respectively. Then,

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 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 values of $a, b$ and $c$ are