Write 'True' or 'False' and justify your answer:
The capacity of a cylindrical vessel with a hemispherical portion raised upward at the bottom as shown in the figure is $\frac{\pi r^{2}}{3} [3 h-2 r]$.

(TRUE) True.
We know that the capacity (volume) of a cylindrical vessel is given by $V_{cylinder} = \pi r^{2} h$.
The capacity (volume) of a hemisphere is given by $V_{hemisphere} = \frac{2}{3} \pi r^{3}$.
From the figure,the hemispherical portion is raised upward at the bottom,which means it occupies space inside the cylinder. Therefore,the capacity of the vessel is the volume of the cylinder minus the volume of the hemisphere.
Capacity of the vessel = $V_{cylinder} - V_{hemisphere}$
$= \pi r^{2} h - \frac{2}{3} \pi r^{3}$
$= \pi r^{2} (h - \frac{2}{3} r)$
$= \frac{\pi r^{2}}{3} (3h - 2r)$.