$A$ radioactive element has a rate of disintegration of $16,000 \text{ disintegrations per minute}$ at a particular instant. After $4 \text{ minutes}$, it becomes $2000 \text{ disintegrations per minute}$. The decay constant per minute is:

Two radioactive materials $R_1$ and $R_2$ have decay constants $6 \lambda$ and $\lambda$, respectively. The half-life of $R_2$ is $1.4 \times 10^{17} \,s$. Initially, they contain the same number of nuclei. The time at which the ratio of the remaining nuclei of $R_2$ to that of $R_1$ will be $e$ is (Let $\ln 2 = 0.7$):

Two radioactive samples $A$ and $B$ have half-lives $T_1$ and $T_2$ $(T_1 > T_2)$ respectively. At $t=0$,the activity of $B$ was twice the activity of $A$. Their activity will become equal after a time:

An element $X$ with a half-life of $1.4 \times 10^9$ years decays to form another stable element $Y$. $A$ sample is taken from a rock that contains both $X$ and $Y$ in the ratio $1:7$. If at the time of formation of the rock,$Y$ was not present in the sample,then the age of the rock in years is

The decay constant of radium is $\lambda$. By a suitable process,its compound radium bromide is obtained. The decay constant of radium bromide will be:

For a radioactive element,the mean life is $\tau$. At time $t = 0$,the number of nuclei decaying per unit time is $n$. The number of nuclei that have decayed between time $0$ and $t$ is: