For a substance,the average life for $\alpha$-emission is $1620 \ years$ and for $\beta$-emission is $405 \ years$. After how much time will $\frac{1}{4}$ of the material remain due to simultaneous emission?

Two radioactive nuclei $A$ and $B$ both convert into a stable nucleus $C$. At time $t = 0$,the number of nuclei of $A$ is $4N_0$ and that of $B$ is $N_0$. The half-life of $A$ is $1 \, min$ and that of $B$ is $2 \, min$. Initially,the number of nuclei of $C$ is zero. At what time are the rates of disintegration of $A$ and $B$ equal?

After $40 \, days$,the $1/16$th part of a radioactive element remains undecayed. What is its half-life (in $, days$)?

The graph shows the $\log$ of activity $\log R$ of a radioactive material as a function of time $t$ in minutes. The half-life (in minutes) for the decay is closest to

$A$ radioactive material decays by simultaneous emission of two particles with respective half-lives $1620$ years and $810$ years. The time (in years) after which one-fourth of the material remains is:

$A$ 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