Magnetic characteristics of epitaxial Ni1-xCox(001) (x=0, 0.16, and 0.50) films with nominal 200 nm thickness on Cu(001)/Si(100) substrates have been investigated by magnetization and ferromagnetic resonance measurements in order to better clarify the rationale for the large variation in the magnetic exchange stiffness constant A, previously determined from different measurements. The exchange constant as well as the saturation magnetization, effective demagnetizing field, fourth-order magnetocrystalline, and second-order perpendicular uniaxial magnetic anisotropy fields has been determined. The analyses of low-temperature saturation magnetization data on these films yield A values that increase from 0.82×10-6erg/cm for a pure Ni film to 2.27×10-6erg/cm for the Ni0.50Co0.50 film. Furthermore, spin-wave resonance volume modes observed in x=0 and 0.16 films indicate that the surface plays a role in the exchange stiffness constant determination as the surface anisotropy constants are found to be approximately 1 and 4 erg/cm2, respectively. The latter value is substantially larger than that for any other system reported so far.