In an old rock,the ratio of uranium to lead nuclei is $1:1$. The half-life of uranium is $4.5 \times 10^9$ years. If the rock initially contained only uranium nuclei,how old is the rock?

$A$ radioactive element having a half-life of $30 \ min$ is undergoing beta decay. The fraction of the radioactive element that remains undecayed after $90 \ min$ 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

Samples of two radioactive nuclides,$X$ and $Y$,each have equal activity $A_0$ at time $t = 0$. $X$ has a half-life of $24$ years and $Y$ has a half-life of $16$ years. The samples are mixed together. What will be the total activity of the mixture at $t = 48$ years?

Two radioactive elements $R$ and $S$ disintegrate as:
$R \rightarrow P + \alpha; \lambda_R = 4.5 \times 10^{-3} \, \text{years}^{-1}$
$S \rightarrow P + \beta; \lambda_S = 3 \times 10^{-3} \, \text{years}^{-1}$
Starting with the number of atoms of $R$ and $S$ in the ratio of $2:1$,what will be this ratio after the lapse of three half-lives of $R$?

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: