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. . . . . . .

After two hours, one- sixteenth of the starting amount of a certain radioactive isotope remained undecayed. The half life of the isotope is

At a given instant, say $t = 0,$ two radioactive substances $A$ and $B$ have equal activates. The ratio $\frac{{{R_B}}}{{{R_A}}}$ of their activities. The ratio $\frac{{{R_B}}}{{{R_A}}}$ of their activates after time $t$ itself decays with time $t$ as $e^{-3t}.$ If the half-life of $A$ is $ln2,$ the half-life of $B$ is

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

A parent nucleus $X$ is decaying into daughter nucleus $Y$ which in turn decays to $Z$. The half lives of $X$ and $Y$ are $40000 \,yr$ and $20 \,yr$, respectively. In a certain sample, it is found that the number of $Y$ nuclei hardly changes with time. If the number of $X$ nuclei in the sample is $4 \times 10^{20}$, the number of $Y$ nuclei present in it is