The half-life of a radioactive isotope $X$ is $20$ years. It decays to another element $Y$,which is stable. The two elements $X$ and $Y$ were found to be in the ratio $1:7$ in a sample of a given rock. The age of the rock is estimated to be:

$A$ and $B$ are two radioactive elements. The mixture of these elements shows a total activity of $1200 \text{ disintegrations/minute}$. The half-life of $A$ is $1 \text{ day}$ and that of $B$ is $2 \text{ days}$. What will be the total activity after $4 \text{ days}$? Given,the initial number of atoms in $A$ and $B$ are equal.

The mean lives of a radioactive sample are $30 \, years$ and $60 \, years$ for $\alpha$-emission and $\beta$-emission,respectively. If the sample decays by both $\alpha$-emission and $\beta$-emission simultaneously,then the time after which only one-fourth of the sample remains is approximately ............ $years$.

$A$ freshly prepared radioactive source of half-life $2$ hours $30$ minutes emits radiation which is $64$ times the permissible safe level. The minimum time,after which it would be possible to work safely with the source,will be in hours.

The activity of a radioactive sample is measured as $N_0$ counts per minute at $t = 0$ and $N_0/e$ counts per minute at $t = 5 \, minutes$. The time (in $minutes$) at which the activity reduces to half its value is

The ratio of the number of active nuclei of two different radioactive samples is $2:3$. Their half-lives are $1 \ h$ and $2 \ h$ respectively. The ratio of the number of active nuclei after $6 \ h$ will be: