We apply strong magnetic fields of H=28.5 to 43 T to suppress superconductivity (SC) in the cuprates Bi2Sr2-xLaxCuO6+delta (x=0.65, 0.40, 0.25, 0.15, and 0), and investigate the low temperature (T) normal state by Cu-63 nuclear spin-lattice relaxation rate (1/T-1) measurements. We find that the pseudogap (PG) phase persists deep inside the overdoped region but terminates at xsimilar to0.05, which corresponds to the hole doping concentration of approximately 0.21. Beyond this critical point, the normal state is a Fermi liquid that persists as the ground state when superconductivity is removed by the magnetic field. A comparison of the superconducting state with the H-induced normal state in the x=0.40 (T-c=32 K) sample indicates that there remains substantial part of the Fermi surface even in the fully developed PG state, which suggests that the PG and SC are coexisting matters.