Bianchi type-III, V and VI0 cosmological models are obtained in the extended theory of gravitation proposed by Harko et al. (2011) as f(R,T)=R+υe−νR+λT, where R is the Ricci scalar, T is the trace of the energy–momentum tensor, υ, ν, and λ are constants. The exact solutions of the field equations of the three models are obtained with help of the hyperbolic scale factor and the proportionality of shear scalar with the expansion scalar. In this work, we have concentrated on ν, which plays a vital role in explaining the evolution of the cosmos. For various values of ν, the behaviour of all the physical properties of the three models are studied with the help of their plots, and they represented the accelerated expansion of the universe. We have derived the geometrical parameters in terms of cosmic time (t) and redshift (z). Also, all the energy conditions have been studied and found that sudden singularity may exist in both Bianchi type-V, VI0 models, whereas in Bianchi type-III, except SEC all the other energy conditions are fulfilled. The behaviour of the r−s plane and r−q plane for various values of the parameter γ have been studied. For the obtained models the stability analysis of the background solutions is discussed through perturbation techniques and observed that the background solutions are quite stable. For all three models, the values of the obtained parameters match the recent cosmological observations.