Hydrodynamic equations for the free expansion of a quasineutral plasma with two-electron temperatures are solved using the self-similarity method. The nature of the solutions over a wide range of tau , the temperature ratio and rho ₀, the initial density ratio of the hot to cold electrons are discussed. For tau < tau *=5+ square root 24 (a threshold value), the ion current profile shows a double peak structure which can be measured experimentally. Subsequent comparison of this with the theoretical formulation is proposed for evaluating the temperatures and initial densities of the electrons. For tau > tau *, the solutions are discontinuous with respect to the self-similarity variable xi signifying the onset of a rarefaction shock (RS). The RS solution predicts the vanishing of ion current in a certain range of ion velocities. For rho ₀> rho * (a critical value) the RS emerges in the ambient plasma itself. Based on these signatures of the shock, a general diagnostic technique is proposed for a laser-produced two-temperature plasma.