Which of the following quantities is dimensionless

The correct option is None of these.

To determine which, if any, of the given quantities is dimensionless, we need to analyze their dimensions step-by-step.

1. Gravitational constant ($G$):
According to Newton's law of gravitation, the force ($F$) between two masses ($m_1$ and $m_2$) separated by a distance ($r$) is given by:

$F=G\frac{m_1{m}_2}{r^2}$

Rearranging the formula to solve for $G$:

$G=\frac{F{r}^2}{m_1{m}_2}$

Substituting the dimensional formulas of force $\left[ML{T}^{-2}\right]$, distance $\left[L\right]$, and mass $\left[M\right]$:

$\left[G\right]=\frac{\left[ML{T}^{-2}\right]\left[L^2\right]}{\left[M\right]\left[M\right]}=\left[M^{-1}{L}^3{T}^{-2}\right]$

Thus, the Gravitational constant is not dimensionless.

2. Planck's constant ($h$):
According to Planck's equation, energy ($E$) is related to frequency ($\nu$) by:

$E=h\nu$

Rearranging the formula to solve for $h$:

$h=\frac{E}{\nu }$

Substituting the dimensional formulas of energy $\left[M{L}^2{T}^{-2}\right]$ and frequency $\left[T^{-1}\right]$:

$\left[h\right]=\frac{\left[M{L}^2{T}^{-2}\right]}{\left[T^{-1}\right]}=\left[M{L}^2{T}^{-1}\right]$

Thus, Planck's constant is not dimensionless.

3. Power of a convex lens ($P$):
The power of a lens is the reciprocal of its focal length ($f$):

$P=\frac{1}{f}$

Since focal length is a distance, its dimension is $\left[L\right]$. Therefore, the dimensional formula for Power is:

$\left[P\right]=\left[L^{-1}\right]$

Thus, the Power of a lens is not dimensionless.

Since none of the quantities listed in the options are dimensionless, the correct choice is None of these.