$A$ radioactive sample decays by $\beta$-emission. In the first $2 \ s$,$n$ $\beta$-particles are emitted,and in the next $2 \ s$,$0.25n$ $\beta$-particles are emitted. The half-life of the radioactive nuclei is ...... $s$.

Activity of a radioactive sample is $R_1$ at a time $t_1$ and $R_2$ at a time $t_2$. Its half-life period is $T$. The number of atoms that have disintegrated in the time interval $(t_2 - t_1)$ is equal to $\frac{n(R_1 - R_2)T}{\ln 4}$. Then '$n$' is equal to

At some instant,a radioactive sample $S_1$ having an activity $5\,\mu Ci$ has twice the number of nuclei as another sample $S_2$ which has an activity of $10\,\mu Ci$. The half-lives of $S_1$ and $S_2$ are:

An alloy is composed of two radioactive materials $A$ and $B$ having equal weight. The half-lives of $A$ and $B$ are $10 \ yrs$ and $20 \ yrs$ respectively. After time $t$, the alloy was found to consist of $(1/e) \ kg$ of $A$ and $1 \ kg$ of $B$. If the atomic weights of $A$ and $B$ are the same, then the value of $t$ is (Assume $\ln 2 = 0.7$):

Two radioactive materials $A_1$ and $A_2$ have decay constants of $10 \lambda_0$ and $\lambda_0$. If initially they have the same number of nuclei,the ratio of the number of their undecayed nuclei will be $(1/e)$ after a time $t$. Find $t$.

$A$ radioactive sample decays at a rate of $5000$ disintegrations/minute at a certain instant. After $5$ minutes, the rate becomes $1250$ disintegrations/minute. The decay constant is ....... (in $\ln 2$)