  Strange quark contributions to the neutral current reaction in the neutrino
scattering are investigated on the nucleon level and extended to the $^{12}$C
target nucleus through the neutrino-induced knocked-out nucleon process in the
quasi-elastic region within the framework of a relativistic single particle
model. The incident energy range between 500 MeV and 1.0 GeV is used for the
neutrino(antineutrino) scattering. Effects of the final state interaction for
the knocked-out nucleon are included by a relativistic optical potential. We
found that the sensitivity of the strange quark contents could be salient on
the asymmetry between neutrino and antineutrino scattering cross sections. In
specific, $A (\nu ({\bar \nu}), \nu^{'} ({\bar \nu}^{'}) N)$ reaction is shown
to be very sensitive test in the searches of the strangeness.
