Recently, a model of opinion formation with kinetic exchanges has been proposed in which a spontaneous symmetry-breaking transition was reported [M. Lallouache, A. S. Chakrabarti, A. Chakraborti, and B. K. Chakrabarti, Phys. Rev. E 82, 056112 (2010)]. We generalize the model to incorporate two parameters: λ, to represent conviction, and μ, to represent the influencing ability of individuals. A phase boundary given by λ=1-μ/2 is obtained separating the symmetric and symmetry broken phases: The effect of the influencing term enhances the possibility of reaching a consensus in the society. The time scale diverges near the phase boundary in a power-law manner. The order parameter and the condensate also show power-law growth close to the phase boundary albeit with different exponents. The exponents in general change along the phase boundary, indicating a nonuniversality. The relaxation times, however, become constant with increasing system size near the phase boundary, indicating the absence of any diverging length scale. Consistently, the fluctuations remain finite but show strong dependence on the trajectory along which it is estimated.