  The neutral-current neutrino-nucleus scattering is calculated through the
neutrino-induced knocked-out nucleon process in the quasielastic region by
using a relativistic single particle model for the bound and continuum states.
The incident energy range between 500 MeV and 1.0 GeV is used for the neutrino
(antineutrino) scattering on ^{12}C target nucleus. The effects of the final
state interaction of the knocked-out nucleon are studied not only on the cross
section but also on the asymmetry due to the difference between neutrinos and
antineutrinos, within a relativistic optical potential. We also investigate the
sensitivity of the strange quark contents in the nucleon on the asymmetry.
