The radioactivity of a certain radioactive element drops to $\frac{1}{64}$ of its initial value in $30$ seconds. Its half-life is ............. seconds.

If $75 \%$ of a radioactive sample disintegrates in $16 \text{ days}$, the half-life of the radioactive sample is (in $\text{ days}$)

$A$ radioactive sample has ${N_0}$ active atoms at $t = 0$. If the rate of disintegration at any time is $R$ and the number of atoms is $N$,then the ratio $R/N$ varies with time as:

An archaeologist analyses the wood in a prehistoric structure and finds that the ratio of $C^{14}$ (half-life $= 5700 \, years$) to $C^{12}$ is only one-fourth of that found in the cells of buried plants. The age of the wood is about .......... $years$.

Following statements related to radioactivity are given below:
$(A)$ Radioactivity is a random and spontaneous process and is dependent on physical and chemical conditions.
$(B)$ The number of undecayed nuclei in the radioactive sample decays exponentially with time.
$(C)$ Slope of the graph of $\log_{e}$ (no. of undecayed nuclei) $Vs.$ time represents the negative reciprocal of mean life time $(-\frac{1}{\tau})$.
$(D)$ Product of decay constant $(\lambda)$ and half-life time $(T_{1/2})$ is constant,equal to $\ln(2)$.
Choose the most appropriate answer from the options given below.

The activity of a radioactive sample is $I_0$ counts/minute at $t = 0$ and $I_0/e$ counts/minute at $t = 5$ minutes. At what time (in minutes) will its activity decrease to half of its initial value?