Q.Which of the following pairs of physical quantities have the same dimensions ?

Every derived quantity reduces to some combination of powers of the seven fundamental (base) quantities -- length $[L]$, mass $[M]$, time $[T]$, electric current $[A]$, temperature $[K]$, amount of substance $[\text{mol}]$, luminous intensity $[\text{cd}]$ -- and this combination, written in square brackets, is called the quantity's dimensional formula. Writing that formula as an equation (e.g. $[v]=[M^0LT^{-1}]$) is the dimensional equation.

Worked derivation (velocity): $\text{velocity}=\dfrac{\text{displacement}}{\text{time}}=\dfrac{[L]}{[T]}=[M^0LT^{-1}]$ -- dimension $0$ in mass, $1$ in length, $-1$ in time.

Dimensional formulas of common quantities follow directly from their defining relation: force $=$ mass $\times$ acceleration $=[MLT^{-2}]$; work $=$ force $\times$ distance $=[ML^2T^{-2}]$; Planck's constant $h=$ energy/frequency $=[ML^2T^{-2}]/[T^{-1}]=[ML^2T^{-1}]$; the gravitational constant $G$ (from Newton's law $F=Gm_1m_2/r^2$) $=[\text{force}\times\text{distance}^2]/\text{mass}^2=[M^{-1}L^3T^{-2}]$; torque, being force $\times$ distance just like work, shares work's dimensional formula $[ML^2T^{-2}]$ -- so torque and energy, despite measuring physically different things, have the same dimensions.

Classifying by dimension. Every physical quantity falls into exactly one of four boxes:

1. Dimensional variables -- have dimensions, and take variable values (length, velocity, acceleration).
2. Dimensionless variables -- no dimensions, but variable values (specific gravity, strain, refractive index). …