For a radioactive material, half-life is $10$ minutes. If initially there are $600$ number of nuclei, the time taken (in minutes) for the disintegration of $450$ nuclei is:

A radio isotope $X$ with a half-life $1.4\times {10}^9$ years decays to $Y$ which is stable. A sample of the rock from a cave was found to contain $X$ and $Y$ in the ratio $1:7$. The age of the rock is:

Half-lives of two radioactive elements $A$ and $B$ are $20$ minutes and $40$ minutes, respectively. Initially, the samples have equal number of nuclei. After $80$ minutes, the ratio of decayed numbers of $A$ and $B$ nuclei will be:

The half-life of radioactive radon is $3.8$ days. The time at the end of which $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$20$}\right.$th of the radon sample will remain undecayed is:

(Given: ${\log }_{10} e=0.4343$)

The half-life period of a radioactive element $X$ is same as the mean-life time of another radioactive element $Y$. Initially, both of them have the same number of atoms. Then: