Summary: The dielectric tensor, modified plasma dispersion function and dispersion relation for Whistler mode instability in an infinite magnetoplasma are obtained in the case of cold plasma injection to background hot anisotropic generalized bi-Lorentzian (\(\kappa\)) plasma in the presence of external perpendicular a.c. electric field. The method of characteristics solutions using perturbed and unperturbed particle trajectories have been used to determine the perturbed distribution function. Integrals and modified plasma dispersion function \(Z^{*}_{\kappa}(\xi)\) are reduced in power series expansion form. Numerical methods using computer techniques have been used to obtain a temporal growth rate for magnetospheric plasma at geostationary height. The bi-Lorentzian (\(\kappa\)) plasma is reducible to various forms of distribution function by changing the spectral index \(\kappa\). The results of bi-Lorentzian (\(\kappa\)) plasma are compared with those of bi-Maxwellian plasma. It has been found that the addition of cold plasma injection gives different frequency spectra. The a.c. frequency of moderate amplitude increases the growth rate and instability in \(\overline{\mathbf K}\) space to lower range. Growth rate maximum is not affected by a.c. frequencies. However, it shifts the maximum to lower \(\overline{\mathbf K}\) space in both cases, rather than on the variation of the magnitude. Thus a physical situation like this may explain emission of various high frequency whistler emissions by cold plasma injection. The potential application of controlled plasma experiments in the laboratory and for planetary atmosphere are indicated.