Write True or False and justify your answer in each of the following:
If the radius of a cylinder is doubled and its curved surface area is not changed,the height must be halved.

(A) Let $r$ be the radius and $h$ be the height of the cylinder.
The formula for the curved surface area $(CSA)$ of a cylinder is $CSA = 2 \pi r h$.
According to the problem,the new radius $r' = 2r$ and the new height $h' = h/2$.
Calculating the new curved surface area $(CSA')$:
$CSA' = 2 \pi r' h' = 2 \pi (2r) (h/2) = 2 \pi r h$.
Since $CSA' = CSA$,the curved surface area remains unchanged when the radius is doubled and the height is halved.
Therefore,the given statement is True.