The half-life of a radioactive isotope is $30 \,h$. How long will it take to get reduced to $12.5 \%$ of its initial amount (in $\,h$)?

The relation between half-life $(T)$ and decay constant $(\lambda)$ is

Using a nuclear counter, the count rate of emitted particles from a radioactive source is measured. At $t = 0$, it was $1600$ counts per second, and at $t = 8 \, s$, it was $100$ counts per second. The count rate observed, in counts per second, at $t = 6 \, s$ is close to:

$A$ radioactive sample decays by two modes: $\alpha$-decay and $\beta$-decay. $66.6\%$ of the time it decays by $\alpha$-decay and $33.3\%$ of the time it decays by $\beta$-decay. If the effective half-life of the sample is $60 \text{ years}$,what will be the half-life of the sample if it decays only by $\alpha$-decay? (in years)

The half-life period of a radioactive element $A$ is $62 \text{ years}$. It decays into another stable element $B$. An archaeologist found a sample in which $A$ and $B$ are in $1 : 15$ ratio. The age of the sample is (in $\text{ years}$)

The activity of a sample of radioactive material is $A_1$ at time $t_1$ and $A_2$ at time $t_2$ $(t_2 > t_1)$. If its mean life is $T$,then which of the following is true?