During mean life of a radioactive element, the fraction that disintegrates is

The half-life period of a radio-active element $X$ is same as the mean life time of another radio-active element $Y$ Initially they have the same number of atoms. Then

Consider two nuclei of the same radioactive nuclide. One of the nuclei was created in a supernova explosion $5$ billion years ago. The other was created in a nuclear reactor $5$ minutes ago. The probability of decay during the next time is

In a radioactive decay process, the activity is defined as $A=-\frac{\mathrm{d} N}{\mathrm{~d} t}$, where $N(t)$ is the number of radioactive nuclei at time $t$. Two radioactive sources, $S_1$ and $S_2$ have same activity at time $t=0$. At a later time, the activities of $S_1$ and $S_2$ are $A_1$ and $A_2$, respectively. When $S_1$ and $S_2$ have just completed their $3^{\text {rd }}$ and $7^{\text {th }}$ half-lives, respectively, the ratio $A_1 / A_2$ is. . . . . . .

A radioactive sample has ${N_0}$ active atoms at $t = 0$. If the rate of disintegration at any time is $R$ and the number of atoms is $N$, then the ratio $ R/N$ varies with time as

Carbon $ - 14$ decays with half-life of about $5,800\, years$. In a sample of bone, the ratio of carbon $ - 14$ to carbon $ - 12$ is found to be $\frac{1}{4}$ of what it is in free air. This bone may belong to a period about $x$ centuries ago, where $x$ is nearest to