$A$ sample of a radioactive element contains $4 \times 10^{16}$ active nuclei. If the half-life of the element is $10$ days,then the number of decayed nuclei after $30$ days is ........ $\times 10^{16}$.

$A$ radioactive sample decays by two modes: $\alpha$-decay and $\beta$-decay. $66.6 \%$ of the time it decays by $\alpha$-decay and $33.3 \%$ of the time it decays by $\beta$-decay. If the half-life of the sample is $60 \text{ years}$,what will be the half-life of the sample if it decays only by $\alpha$-decay?

If the half-life of a radioactive material is $10 \ years$,then the percentage of the material decayed in $30 \ years$ is (in $\%$)

Two radioactive materials $X_1$ and $X_2$ have decay constants $5 \lambda$ and $\lambda$ respectively. Initially,they have the same number of nuclei. After time $t$,the ratio of the number of nuclei of $X_1$ to that of $X_2$ is $\frac{1}{e}$. Then $t$ is equal to:

The half-life of a sample of a radioactive substance is $1 \text{ hour}$. If $8 \times 10^{10}$ atoms are present at $t = 0$,then the number of atoms decayed in the duration $t = 2 \text{ hours}$ to $t = 4 \text{ hours}$ will be

The average life $T$ and the decay constant $\lambda$ of a radioactive nucleus are related as