Two radioactive materials $A$ and $B$ have decay constants $25 \lambda$ and $16 \lambda$ respectively. If initially they have the same number of nuclei,then the ratio of the number of nuclei of $B$ to that of $A$ will be $e$ after a time $t = \frac{1}{a \lambda}$. The value of $a$ is $......$

The half-life of radioactive Radon is $3.8 \ days$. The time at the end of which $1/20^{th}$ of the Radon sample will remain undecayed is ........... $days$ (Given $\log_{10} e = 0.4343$).

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

The half-life of a radioactive element is $12.5 \; h$ and its initial quantity is $256 \; g$. After how much time (in $h$) will its quantity remain $1 \; g$?

Two radioactive isotopes $P$ and $Q$ have half-lives $10 \text{ minutes}$ and $15 \text{ minutes}$ respectively. Freshly prepared samples of each isotope initially contain the same number of atoms. After $30 \text{ minutes}$,the ratio $\frac{\text{number of atoms of } P}{\text{number of atoms of } Q}$ will be

If the half-life of a radioactive sample is $10\, hours$,its mean life is .......... $hours$.