$A$ radioactive substance of half-life $138.6 \text{ days}$ is placed in a box. After $n$ days, only $20\%$ of the substance is present. Then the value of $n$ is $[\ln(5) = 1.61]$.

$A$ heavy nucleus $Q$ of half-life $20 \text{ minutes}$ undergoes alpha-decay with a probability of $60 \%$ and beta-decay with a probability of $40 \%$. Initially,the number of $Q$ nuclei is $1000$. The number of alpha-decays of $Q$ in the first one hour is:

The activity of an element becomes $\frac{1}{64}$ of its original value in $60 \ s$. Then the half-life period is ............ $s$.

$A$ radioactive substance contains $10000$ nuclei and its half-life period is $20$ days. The number of nuclei present at the end of $10$ days is

$A$ radioactive nucleus can decay by two different processes. The half-life for the first process is $3.0 \, hours$ while it is $4.5 \, hours$ for the second process. The effective half-life of the nucleus will be $......... \, hours.$

At a given instant,a radioactive sample has $N$ active nuclei. If its decay constant is $\lambda$,which of the following relations is incorrect? (Note: $\lambda$ is very small.)