$A$ radioactive element $A$ converts into another stable element $B$. The half-life of $A$ is $1.5 \ hrs$. After time $t$,the ratio of the number of atoms of $A$ to $B$ is found to be $1:8$. Then $t$ in hours is:

The half-life $(T)$ and the disintegration constant $(\lambda)$ of a radioactive substance are related as:

The half-life of $^{215}At$ is $100 \mu s$. The time taken for the radioactivity of a sample of $^{215}At$ to decay to $\frac{1}{16}$ of its initial value is .........$\mu s$.

$A$ radioactive source has a half-life of $6 \,h$. $A$ freshly prepared sample of the same exhibits radioactivity $32$ times the permissible safe value. The minimum time after which it would be possible to work safely with the source is (in $\,h$)

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 half-life of a radioactive sample is $20 \ days$. This means that: