At any instant, two elements $X _1$ and $X _2$ have same number of radioactive atoms. If the decay constant of $X _1$ and $X _2$ are $10 \lambda$ and $\lambda$ respectively. then the time when the ratio of their atoms becomes $\frac{1}{e}$ respectively will be

In a radioactive decay chain, ${ }_{90}^{232} Th$ nucleus decays to ${ }_{82}^{212} Pb$ nucleus. Let $N _\alpha$ and $N _\beta$ be the number of $\alpha$ and $\beta^{-}$particles, respectively, emitted in this decay process. Which of the following statements is (are) true?

$(A)$ $N _\alpha=5$  $(B)$ $N _\alpha=6$  $(C)$ $N _\beta=2$  $(D)$ $N _\beta=4$

If $T$ is the half life of a radioactive material, then the fraction that would remain after a time $\frac{T}{2}$ is

A radioactive substance is being produced at a constant rate of $10\, nuclei/s.$ The decay constant of the substance is $1/2\, sec^{-1}.$ After what time the number of radioactive nuclei will become $10$ $?$ Initially there are no nuclei present. Assume decay law holds for the sample.

The decay constant of a radioactive element is $0.01$ per second. Its half life period is …….$sec$