Zero-and longitudinal-field muon-spin-relaxation ͑LF-SR͒ measurements were carried out on the randomness-introduced quantum spin system Tl 1−x K x CuCl 3 . The relative temperature change in the muon-spinrelaxation rate in longitudinal-fields, which corresponds to the wave-vector integration of the generalized dynamical susceptibility, was deduced from LF-SR measurements. Peak structure in the relative temperature change of for x = 0.60 is observed at T ϳ 3 K in 3950 G, and the temperature where the peak is observed decreases with decreasing the magnetic field. This behavior is interpreted as the soft mode of spin waves in the vicinity of the quantum critical point.