A 12.3 eV electron beam is used to bombard gaseous hydrogen at room temperature. Upto which energy level the hydrogen atoms would be excited?

The hydrogen atoms should be excited to n=3 as energy difference between n=1 and n=3 is 12.08 eV.

hc = 1240 eVnm
For second wavelength of Balmer series:
hc/λ₂ = (13.6eV)(1/2² - 1/3²)
1240 eV.nm/λ₂ = 13.6 × 5/36
λ₂ = 1240 × 36 / (13.6 × 5) nm = 656.2 nm = 6562 Å

For second wavelength of Lyman series:
hc/λ₂' = (13.6eV)(1/1² - 1/2²)
1240 eVnm/λ₂' = 13.6 × 3/4
λ₂' = 1240 × 4 / (13.6 × 3) nm = 121.5 nm = 1215 Å