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 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 ..........$min$

A piece of wood from the ruins of an ancient building was found to have a $^{14}C$ activity of $12$ disintegrations per minute per gram of its carbon content. The $^{14}C$ activity of the living wood is $16$ disintegrations per minute per gram. How long ago did the tree, from which the wooden sample came, die? Given half-life of $^{14}C$ is $5760$ years.

$3.8$ days is the half-life period of a sample. After how many days, the sample will become $\frac{{1}}{{8}} \, th$ of the original substance

Two radioactive nuclei $P$ and $Q,$ in a given sample decay into a stable nucleus $R.$ At time $t = 0,$ number of $P$ species are $4\,\, N_0$ and that of $Q$ are $N_0$. Half-life of $P$ (for conversion to $R$) is $1$ minute where as that of $Q$ is $2$ minutes. Initially there are no nuclei of $R$ present in the sample. When number of nuclei of $P$ and $Q$ are equal, the number of nuclei of $R$ present in the sample would be