Radioactive material $A$ has decay constant $8 \lambda$ and material $B$ has decay constant $\lambda$. Initially,they have the same number of nuclei. After what time will the ratio of the number of nuclei of material $B$ to that of $A$ be $\frac{1}{e}$?

$A$ radioactive isotope $X$ with a half-life of $1.37 \times 10^9$ years decays to $Y$,which is stable. $A$ sample of rock from the moon was found to contain both the elements $X$ and $Y$ in the ratio of $1 : 7$. The age of the rock is

The relation between the mean life time $\tau$ and the half-life time $T_{1/2}$ of a radioactive substance is:

The normal activity of living carbon-containing matter is found to be about $15$ decays per minute for every gram of carbon. This activity arises from the small proportion of radioactive $_{6}^{14}C$ present with the stable carbon isotope $_{6}^{12}C$. When the organism is dead,its interaction with the atmosphere (which maintains the above equilibrium activity) ceases and its activity begins to drop. From the known half-life ($5730$ years) of $_{6}^{14}C$,and the measured activity,the age of the specimen can be approximately estimated. This is the principle of $_{6}^{14}C$ dating used in archaeology. Suppose a specimen from Mohenjodaro gives an activity of $9$ decays per minute per gram of carbon. Estimate the approximate age (in years) of the Indus-Valley civilisation.

$A$ radio-isotope has a half-life of $5$ years. The fraction of the atoms of this material that would decay in $15$ years will be

Plutonium decays with a half-life of $24000 \, \text{years}$. If the plutonium is stored for $72000 \, \text{years}$, then the fraction of plutonium that remains is