To solve this problem, we will use Torricelli's theorem, which relates the speed of efflux of a fluid under gravity to the height of the fluid column.$\\$According to Torricelli's theorem, the speed $v$ of efflux is given by:$\\ v = \sqrt{2gh} \\$where $g = 10$ m/s$^2$ is the acceleration due to gravity and $h = 2$ m is the height of the fluid column.$\\$Substitute the known values:$\\ v = \sqrt{2 \times 10 \times 2} \\ v = \sqrt{40} \\ v = 6.32$ m/s$\\$The rate of flow of water (volume flow rate) $Q$ through the hole is given by:$\\ Q = A \cdot v \\$where $A$ is the area of the cross-section of the hole. Given $A = 2$ mm$^2,$ we need to convert this to m$^2: \\ A = 2 \times 10^{-6}$ m$^2 \\$Substitute the values into the equation for $Q: \\ Q = (2 \times 10^{-6}) \cdot 6.32 \\ Q = 12.64 \times 10^{-6}$ m$^3/$s$\\$Rounding to one decimal place, the rate of flow of water is approximately:$\\ Q \approx 12.6 \times 10^{-6}$ m$^3/$s$\\$Therefore, the correct option is Option 1.