The activity of a sample of a radioactive material is ${A_1}$ at time ${t_1}$ and ${A_2}$ at time ${t_2}$ $({t_2} > {t_1})$. If its mean life is $T$,then:

Two radioactive samples $A$ and $B$ have half-lives $T_1$ and $T_2$ $(T_1 > T_2)$ respectively. At $t=0$,the activity of $B$ was twice the activity of $A$. Their activity will become equal after a time:

Two radioactive materials $A_1$ and $A_2$ have decay constants of $10 \lambda_0$ and $\lambda_0$. If initially they have the same number of nuclei,the ratio of the number of their undecayed nuclei will be $(1/e)$ after a time $t$. Find $t$.

The half-life of a radioactive substance is $20 \ min$. The time taken for the decay to increase from $20\%$ to $80\%$ is ........ $min$.

If a radioactive substance reduces to $\frac{1}{16}$ of its original mass in $40$ days,what is its half-life in days?

${ }_{92}^{238} U$ is known to undergo radioactive decay to form ${ }_{82}^{206} Pb$ by emitting alpha and beta particles. $A$ rock initially contained $68 \times 10^{-6} \text{ g}$ of ${ }_{92}^{238} U$. If the number of alpha particles that it would emit during its radioactive decay of ${ }_{92}^{238} U$ to ${ }_{82}^{206} Pb$ in three half-lives is $Z \times 10^{18}$,then what is the value of $Z$?