  We study the random field $p$-spin model with Ising spins on a fully
connected graph using the theory of large deviations in this paper. This is a
good model to study the effect of quenched random field on systems which have a
sharp first order transition in the pure state. For $p=2$, the phase-diagram of
the model, for bimodal distribution of the random field, has been well studied
and is known to undergo a continuous transition for lower values of the random
field ($h$) and a first order transition beyond a threshold, $h_{tp}(\approx
0.439)$. We find the phase diagram of the model, for all $p \ge 2$, with
bimodal random field distribution, using large deviation techniques. We also
look at the fluctuations in the system by calculating the magnetic
susceptibility. For $p=2$, beyond the tri-critical point in the regime of first
order transition, we find that for $h_{tp}<h<0.447$, magnetic susceptibility
increases rapidly (even though it never diverges) as one approaches the
transition from the high temperature side. On the other hand, for $0.447<h \le
0.5$, the high temperature behaviour is well described by the Curie-Weiss law.
For all $p \ge 2$, we find that for larger magnitudes of the random field
($h>h_o=1/p!$), the system does not show ferromagnetic order even at zero
temperature. We find that the magnetic susceptibility for $p \ge 3$ is
discontinuous at the transition point for $h<h_o$.
