The half-life of a radioactive isotope $X$ is $50$ years. It decays to another element $Y$ which is stable. The two elements $X$ and $Y$ were found to be in the ratio of $1 : 15$ in a sample of a given rock. The age of the rock was estimated to be..........$years$.

There are three lumps of a given radioactive substance. Their activity is in the ratio of $1 : 2 : 3$ now. What will be the ratio of their activities at any further date?

$A$ radioactive element which can decay by two processes has half-life $t_1$ for the first process and half-life $t_2$ for the second process. Let $\langle t \rangle$ be the effective average-life of this element. Which of the following is correct?

The half-life of a radioactive isotope $X$ is $1.37 \times 10^9 \, y$. It decays to a stable isotope $Y$. In a rock sample from the moon,the ratio of $X$ to $Y$ is $1:7$. The age of the rock is:

Two radioactive isotopes $P$ and $Q$ have half-lives $10 \text{ minutes}$ and $15 \text{ minutes}$ respectively. Freshly prepared samples of each isotope initially contain the same number of atoms. After $30 \text{ minutes}$,the ratio $\frac{\text{number of atoms of } P}{\text{number of atoms of } Q}$ will be

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 $\ln 2 = 0.7$):