The family lives in the AIIMS residential complex in Phulwarisharif area of Patna.
The reasons for the suicide is unclear say police sources.
The police are trying to find out how the teenager got the pistol, which reportedly has no licence. The boy's parents told the police that they did not have any firearms.
The police have started an investigation into the case.
Sakshi Eknath Bendkule, a resident of Panchavati hanged herself in her house, hours after Maharashtra Board Class 10 results were declared, a police official said.
As per the statistics released by National Crime Records Bureau (NCRB) for the year 2015, almost 8,952 students committed suicide in India. Of the total suicides in India in 2015, 2% of the total was cited as due to failure in examination.