The graph represents the decay of a newly prepared sample of radioactive nuclide $X$ to a stable nuclide $Y$. The half-life of $X$ is $\tau$. The growth curve for $Y$ intersects the decay curve for $X$ after time $T$. What is the time $T$?

${ }_{92}^{238} U$ 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 $Z$?

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 $T$. The time taken for all the nuclei to disintegrate will be

$A$ radioactive source has a half-life of $6 \,h$. $A$ freshly prepared sample of the same exhibits radioactivity $32$ times the permissible safe value. The minimum time after which it would be possible to work safely with the source is (in $\,h$)

The half-life of the isotope $^{11}Na^{24}$ is $15 \text{ hr}$. How much time does it take for $(7/8)$ of a sample of this isotope to decay (in $\text{ hour}$)?