We investigate the capture rate of the cosmic neutrino background on tritium within the Standard Model, extended to incorporate three right-handed singlet neutrinos with explicit lepton-number violation. We consider a scenario where the neutrino mixing matrix factorizes into three independent pairs and analyze the states produced from weak interactions just before neutrino decoupling. Taking into account the unrestricted Majorana mass scale associated with lepton number violation, spanning from the Grand Unification scale to Planck-suppressed values, we observe a gradual transition in the capture rate from a purely Majorana neutrino to a purely (pseudo) Dirac neutrino. We demonstrate that the capture rate is modified if the lightest active neutrino is relativistic, and this can be used to constrain the tiniest value of mass-squared difference , between the active-sterile pair, probed so far. Consequently, the cosmic neutrino capture rate could become a promising probe for discerning the underlying mechanism responsible for generating neutrino masses.