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, $A ( u ({bar u}), u^{} ({bar u}^{}) N)$ reaction is shown to be very sensitive test in the searches of the strangeness.
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