The Bethe lattice approach is used to examine the effect of a random crystal field (RCF) on the mixed spin-1/2 and 5/2 Blume-Capel model. A bimodal form of RCF is considered which either turns the crystal field on or off for given probabilities p and 1 - p, respectively, for the sites with spin-5/2. The exact recursion relations are employed to obtain all the characteristics of the model for given coordination numbers q = 3,4,5 and 6. The thermal variations of the order-parameters are studied and net magnetizations are classified according to the usual letter coding. The phase diagrams are obtained on the probability-temperature and crystal field-temperature planes. It is found that the model yields only one compensation temperature for appropriate values of system parameters.