In this work, we have examined the thermal variations of the order parameters to calculate the phase diagrams for the mixed spin-1/2 and 5/2 Blume-Capel (BC) model on the Bethe lattice (BL) for the bimodal distribution of the bilinear exchange interaction, +/- J, and an adjusting parameter, alpha, between them. In addition, the values of J are distributed on the BL with probabilities p and 1 - p for J > 0 corresponding to the ferromagnetic (FM) phase and J < 0 for the antiferromagnetic (AFM) phase, respectively. The calculations are carried out in terms of the exact recursion relations (ERR) and then the probability distribution is implemented into the model. The phase diagrams are obtained on the (D, T) and (p, T) planes by varying the rest of the parameters for the coordination number q = 3 which corresponds to honeycomb lattice. We have found in addition to the first- and second-order phase transitions, a few critical points such as tricritical and bicritical points.