The orientation of a solute probe in a binary mixture often exhibits multiple relaxation times at the same solvent viscosity but different compositions [Beddard , Nature (London) 294, 145 (1981)]. In order to understand this interesting observation, we have carried out (NPT) molecular dynamics simulation study of rotation of prolate ellipsoids in binary mixtures. The simulations show that for a broad range of model parameters the experimental behavior can be reproduced. The plot of orientational relaxation time versus the solvent viscosity, the latter changed by varying the composition, shows a nonmonotonic viscosity dependence. The nature of the plot is found to depend on the system parameters, especially on the solute-solvent interactions. A mode coupling theoretical analysis of this complex dependence of the rotational relaxation time tau (R) on the binary viscosity eta is presented. The theory can qualitatively explain the origin of the multiple relaxation time at the same viscosity.