True.
Let the radius of a cylinder be r and height be h.
Curved surface area of a cylinder =2πrh
Let the new radius, r1=2r and height be H.
Curved surface of the cylinder is not changed; therefore,
2πrh=2π(2r)×H
⇒2πrh=4πrH
⇒H=2πrh4πr⇒H=h2
Hence, the height must be halved.