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The quasi-elastic scattering of muon neutrino and electrons on a carbontarget are analyzed using the relativistic distorted-wave impulse approximation(RDWIA). We also evaluate the contribution of the two-particle and two-holemeson exchange current ($2p-2h$ MEC) to electroweak response functions. Thenuclear model dependence of the (anti)neutrino cross sections is studied withinthe RDWIA+MEC approach and RDWIA model with the large nucleon axial mass. It isshown that the results for the squared momentum transfer distribution$d\sigma/dQ^2$ and for invariant mass of the final hadronic system distribution$d\sigma/dW$ obtained within these models are substantially different.

Quasi-elastic neutrino charged-current scattering off $^{12}$C: effects of the meson exchange currents and large nucleon axial mass