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Summary form only given. I present results of the detailed asymptotic and numerical results on stability of unit and higher order vortices in coupled Gross-Pitaevskii equations, which model the experimental work. Low lying eigenfrequencies, including ones generating instability of the unit vortex placed in one component of a binary mixture of BEC, have been calculated applying perturbative theory in the limit of the weak interparticle interaction. Results of the perturbative theory have been verified and significantly extended using numerical analysis of the linearised equations for excitations and direct numerical simulation of the coupled Gross-Pitaevskii equations. Two main instability mechanisms have been identified and their dependencies on the relative number of the particles in the condensates and on the strength of the inter-species interaction have been studied. The first mechanism is the drift instability induced by the vortex free component of the condensate and resulting in the motion of the vortex out of the condensate center to its periphery. The second mechanism is the splitting of the higher order vortices into unit ones. Absolute and/or relative increase of the number of particles in the vortex free condensate component has been found to have stabilizing effect on both drift and splitting instabilities. Different scenarios of the splitting instabilities have been found.