JEE Main 2024 — 4 April, Shift 1 — Question 24

If the shortest distance between the lines $\frac{x+2}{2}=\frac{y+3}{3}=\frac{z-5}{4}$ and $\frac{x-3}{1}=\frac{y-2}{-3}=\frac{z+4}{2}$ is $\frac{38}{3 \sqrt{5}} \mathrm{k} \quad$

and $\quad \int_{0}^{\mathrm{k}}\left[\mathrm{x}^{2}\right] \mathrm{dx}=\alpha-\sqrt{\alpha}, \quad$ where $\quad[\mathrm{x}]$ denotes the greatest integer function, then $6 \alpha^{3}$ is equal to $\qquad$