The half-life of a radioactive substance is $T$. The time taken for disintegrating $\frac{7}{8}$th part of its original mass will be

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?

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 ${}_{92}^{238}U$ against $\alpha$-decay is $13.86 \times 10^{16} \,s$. The activity of a $1 \,g$ sample of ${}_{92}^{238}U$ is:

$A$ radioactive substance in a laboratory has a half-life of $2 \text{ hours}$. It is considered safe for laboratory use when its activity reduces to $1/64$ of its initial value. After how many hours will the laboratory be safe?

After $3$ hours,only $0.25 \,mg$ of a pure radioactive material is left. If the initial mass was $2 \,mg$,then the half-life of the substance is ...... $hr$.