The activity of a radioactive sample is $9750$ counts/minute at $t = 0$ and $975$ counts/minute at $t = 5$ minutes. The decay constant is .......... $min^{-1}$.

At a certain moment,the amounts of two radioactive substances are in the ratio $2:1$. Their half-lives are $12 \text{ hours}$ and $16 \text{ hours}$ respectively. What will be the ratio of their remaining amounts after $2 \text{ days}$?

$A$ 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 $\,h$)

$A$ sample of living carbon-containing matter has an activity of $15$ decays/minute per gram of carbon-$14$. $A$ sample from the Indus Valley civilization has an activity of $9$ decays/minute per gram of carbon-$14$. Given the half-life of $C^{14}$ is $5730$ years, find the age of the Indus Valley civilization.

The half-life of a stream of radioactive particles moving along a straight path with a constant kinetic energy of $4 \text{ eV}$ is $1 \text{ minute}$. The percentage of particles which decay before travelling a distance of $3.6 \text{ km}$ is (Mass of the radioactive particles $= 3.2 \times 10^{-21} \text{ kg}$ and charge of the electron $= 1.6 \times 10^{-19} \text{ C}$).

The activity of a radioactive sample is measured as $9750$ counts per minute at $t = 0$ and as $975$ counts per minute at $t = 5$ minutes. The decay constant is approximately ............ per minute.