The activity of a radioactive sample:

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 the isotope $_{11}Na^{24}$ is $15 \, hrs$. How much time does it take for $\frac{7}{8}$ of a sample of this isotope to decay?

The graph represents the decay of a newly prepared sample of radioactive nuclide $X$ to a stable nuclide $Y$. The half-life of $X$ is $\tau$. The growth curve for $Y$ intersects the decay curve for $X$ after time $T$. What is the time $T$?

$1 \, mg$ of gold undergoes decay with a $2.7 \, \text{days}$ half-life period. The amount left after $8.1 \, \text{days}$ is ......... $mg$.

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