$A$ $280\, \text{day}$ old radioactive substance shows an activity of $6000\, \text{dps}$. $140\, \text{days}$ later, its activity becomes $3000\, \text{dps}$. What was its initial activity? ......... $\text{dps}$

The half-life of $^{215}At$ is $100 \,\mu s$. After how much time (in $\mu s$) will $1/16^{th}$ of the sample remain undecayed?

$A$ radioactive nucleus is being produced at a constant rate $\alpha$ per second. Its decay constant is $\lambda$. If $N_0$ is the number of nuclei at time $t = 0$,then the maximum number of nuclei possible is:

Two radioactive materials $X_1$ and $X_2$ contain the same number of nuclei. If $6\lambda \, s^{-1}$ and $4\lambda \, s^{-1}$ are the decay constants of $X_1$ and $X_2$ respectively,the ratio of the number of undecayed nuclei of $X_1$ to that of $X_2$ will be $\left( \frac{1}{e} \right)$ after a time:

The half-life of a radioactive element $A$ is the same as the mean-life of another radioactive element $B.$ Initially, both substances have the same number of atoms, then

The half-life of a radioactive nucleus is $5$ years. The fraction of the original sample that would decay in $15$ years is