At any instant, two elements $X _1$ and $X _2$ have same number of radioactive atoms. If the decay constant of $X _1$ and $X _2$ are $10 \lambda$ and $\lambda$ respectively. then the time when the ratio of their atoms becomes $\frac{1}{e}$ respectively will be

The initial activity of a certain radioactive isotope was measured as $16000\ counts\  min^{-1}$. Given that the only activity measured was due to this isotope and that its activity after $12\, h$ was $2000\ counts\ min^{-1}$, its half-life, in hours, is nearest to

A radioactive element $ThA (_{84}Po^{216})$ can undergo $\alpha$ and $\beta$ are type of disintegrations with half-lives, $T_1$ and $T_2$ respectively. Then the half-life of ThA is

Write and derive the law of radioactivity decay. 

An archaeologist analyses the wood in a prehistoric structure and finds that $C^{14}$ (Half life $= 5700\, years$) to $C^{12}$ is only one-fourth of that found in the cells of buried plants. The age of the wood is about ..........$years$

A count rate meter shows a count of $240$ per minute from a given radioactive source. One hour later the meter shows a count rate of $30$ per minute. The half-life of the source is ..........$min$