Ratio of longest wavelength corresponding to Lyman and Balmer series in hydrogen spectrum is

Half life of radio-active element is 1600 years. The fraction of sample remains undecayed after 6400 years will be

Frequency of the series limit of Balmer series of hydrogen atom in terms of Rydberg's constant (R) and velocity of light (c) is

If the radius of the first Bohr orbit is '$$r$$' then the de-Broglie wavelength of the electron in the $$4^{\text {th }}$$ orbit will be