Two radioactive materials $X_1$ and $X_2$ contain same number of nuclei. If $6\,\lambda {s^{ – 1}}$ and $4\,\lambda {s^{ – 1}}$ are the decay constants of $X_1$ and $X_2$ respectively the ratio of number of nuclei, undecayed of $X_1$ to that of $X_2$ will be $\left( {\frac{1}{e}} \right)$ after a time 

A fraction $f_1$ of a radioactive sample decays in one mean life, and a fraction $f_2$ decays in one half-life.

A $1000 \;MW$ fission reactor consumes half of its fuel in $5.00\; y$. How much $_{92}^{235} U$ (in $kg$) did it contain initially? Assume that the reactor operates $80 \%$ of the time, that all the energy generated arises from the fission of $_{92}^{235} U$ and that this nuclide is consumed only by the fission process.

Two radioactive substances $X$ and $Y$ originally have $N _{1}$ and $N _{2}$ nuclei respectively. Half life of $X$ is half of the half life of $Y$. After three half lives of $Y$, number of nuclei of both are equal. The ratio $\frac{ N _{1}}{ N _{2}}$ will be equal to

The half lives of a radioactive substance are $T$ and $2T$. years for $\alpha  – $ emission and $\beta – $ emission respectively. The total de cay constnnt for simultaneous decay of $\alpha$ and $\beta$ adioactive substance is ___