We present a molecular dynamics simulation study of the liquid-solid transition in a two dimensional system consisting of particles of two different sizes interacting via a truncated Lennard-Jones potential. We work with equal number of particles of each kind and the dispersity Δ in the sizes of the particles is varied by changing the ratio of the particle sizes only. For the monodisperse case (Δ = 0) and for small values of Δ, we find a first order liquid-solid transition on increasing the volume fraction p of the particles. As we increase Δ, the first-order transition coexistence region weakens gradually and completely disappears at high dispersities around Δ = 0.10. At these values of dispersity the high density phase lacks long range translational order but possesses orientational order with a large but finite correlation length. The consequences of this effect of dispersity on the glass transition and on the melting transition in general are discussed.