(a) When a cyclist doubles his speed. His kinetic energy increases four times: KE=12×m×v2
(b) Yes, a body possesses energy even when it is not in motion; Consider a body raised to a certain he ight say h. It its velocity is zero.
Kinetic energy will be zero but the body will have. P.E. = mgh.
Thus, a body may possess energy even though it is not in motion.