The activity of a freshly prepared radioactive sample is $10^{10}$ disintegrations per second,whose mean life is $10^9 \ s$. The mass of an atom of this radioisotope is $10^{-25} \ kg$. The mass (in $mg$) of the radioactive sample is

$A$ sample has a half-life of $10^{33}$ years. If the initial number of nuclei of the sample is $26 \times 10^{24}$,then the number of nuclei decayed in $1$ year is ........... $\times 10^{-7}$.

$A$ radioactive material decays by simultaneous emission of two particles with respective half-lives $1620$ years and $810$ years. The time (in years) after which one-fourth of the material remains is:

An accident in a nuclear laboratory resulted in the deposition of a certain amount of radioactive material with a half-life of $18$ days inside the laboratory. Tests revealed that the radiation level was $64$ times higher than the permissible level required for safe operation. What is the minimum number of days after which the laboratory can be considered safe for use?

The activity of a radioactive sample falls from $700 \; s^{-1}$ to $500 \; s^{-1}$ in $30 \; min$. Its half-life is close to ......... $min$.

The activity of a radioactive substance is $R_1$ at time $t_1$ and $R_2$ at time $t_2$. If $\lambda$ is the decay constant,then which of the following is correct?