An exact calculation of the thermodynamic properties of a linear chain of classical spins with nearest-neighbor bilinear and biquadratic isotropic random exchange interactions is presented. Both open and closed chains are discussed. Only bond disorder is being considered here. It is shown that in contrast to the uniform system, "dipolar disorder points" as well as "quadrupolar disorder points" with qualitative and quantitative differences from the uniform case occur in this system. This is because the competing dipolar and quadrupolar interactions are further enriched by the random signs of these interactions. It is interesting to point out that much of the analysis can be made by means of a study of the eigenvalues of the "transfer kernel." The method can also be applied to obtain exact results for a linear chain of spins of arbitrary dimensionality interacting through an arbitrary isotropic nearest-neighbor Hamiltonian. In the case of the open chain, the Edwards-Anderson expedient is found to be exact in all the cases considered.