The half-life of radium is $1600 \, \text{years}$. The fraction of the radium sample that will remain after $6400 \, \text{years}$ is:

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 $t = 5\, \text{minutes}$. The time (in minutes) at which the activity reduces to half of its initial value is

The half-life of a radioactive substance is $20 \ min$. The time interval between $20\%$ decay and $80\%$ decay is ......... $min$.

If the mass of a radioactive sample is doubled,the activity of the sample and the disintegration constant of the sample are respectively

$A$ sample contains $10^{-2} \ kg$ each of two substances $A$ and $B$ with half-lives $4 \ s$ and $8 \ s$ respectively. The ratio of their atomic weights is $1:2$. The ratio of the amounts (number of atoms) of $A$ and $B$ remaining after $16 \ s$ is $\frac{x}{100}$. The value of $x$ is:

$A$ radioactive nucleus can decay by two different processes. The half-lives of the first and second decay processes are $5 \times 10^3$ years and $10^5$ years respectively. Then, the effective half-life of the nucleus is