Write 'True' or 'False' and justify your answer:
$A$ solid cone of radius $r$ and height $h$ is placed over a solid cylinder having the same base radius and height as that of the cone. The total surface area of the combined solid is $\pi[\sqrt{r^{2}+h^{2}}+3r+2h]$.

(B) False.
The total surface area of the combined solid is the sum of the curved surface area of the cone,the curved surface area of the cylinder,and the area of the bottom base of the cylinder.
$1$. Curved surface area of the cone = $\pi r l$,where slant height $l = \sqrt{r^2 + h^2}$.
$2$. Curved surface area of the cylinder = $2\pi rh$.
$3$. Area of the bottom base of the cylinder = $\pi r^2$.
Total surface area = $\pi r l + 2\pi rh + \pi r^2$
$= \pi r (l + 2h + r)$
$= \pi r [\sqrt{r^2 + h^2} + 2h + r]$.
Since the calculated expression $\pi r [\sqrt{r^2 + h^2} + 2h + r]$ is not equal to the given expression $\pi [\sqrt{r^2 + h^2} + 3r + 2h]$,the statement is False.