$A$ radioactive sample of half-life $ 10 $ days contains $ 1000x $ nuclei. The number of original nuclei present after $ 5 $ days is: (in $x$)

An accident in a nuclear laboratory resulted in the deposition of a certain amount of radioactive material with a half-life of $18$ days inside the laboratory. Tests revealed that the radiation level was $64$ times higher than the permissible level required for safe operation. What is the minimum number of days after which the laboratory can be considered safe for use?

The half-life of a radioactive isotope $X$ is $20$ years. It decays to another element $Y$,which is stable. The two elements $X$ and $Y$ were found to be in the ratio $1:7$ in a sample of a given rock. The age of the rock is estimated to be:

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

$A$ radioactive sample has a half-life of $5$ years. The percentage of the fraction decayed in $10$ years will be (in $\%$)

If the half-life of radium is $77$ days,its decay constant in $day^{-1}$ will be: