At $t = 0$,the counting rate from a radioactive source is $1600 \text{ counts/s}$,and at $t = 8 \text{ s}$,it is $100 \text{ counts/s}$. The counting rate at $t = 6 \text{ s}$ will be:

The half-life period of a radioactive substance is $5\, min$. The amount of substance decayed in $20\, min$ will be..........$\%$

$A$ sample initially contains only $U-238$ isotope of uranium. With time,some of the $U-238$ radioactively decays into $Pb-206$ while the rest of it remains undisintegrated. When the age of the sample is $P \times 10^8$ years,the ratio of the mass of $Pb-206$ to that of $U-238$ in the sample is found to be $7$. The value of $P$ is. . . . . . [Given: Half-life of $U-238$ is $4.5 \times 10^9$ years; $\log_e 2 = 0.693$]

What is the relationship between the decay constant $\lambda$ and the half-life $\tau_{1/2}$?

$A$ parent nucleus $X$ decays into a daughter nucleus $Y$,which in turn decays into $Z$. The half-lives of $X$ and $Y$ are $40000 \, yr$ and $20 \, yr$,respectively. In a certain sample,it is found that the number of $Y$ nuclei hardly changes with time. If the number of $X$ nuclei in the sample is $4 \times 10^{20}$,the number of $Y$ nuclei present in it is

There are $10^{10}$ radioactive nuclei in a given radioactive element. Its half-life time is $1 \text{ minute}$. How many nuclei will remain after $30 \text{ seconds}$? $(\sqrt{2} = 1.414)$