The count rate for $10 \, g$ of radioactive material was measured at different times and this has been shown in the graph. The half-life of the material and the total count in the first half-life period respectively are:

Activity of a radioactive substance is $R_1$ at time $t_1$ and $R_2$ at time $t_2$ $(t_2 > t_1)$. Then the ratio $\frac{R_2}{R_1}$ is:

Given below are two statements:
Statement $I$: The law of radioactive decay states that the number of nuclei undergoing the decay per unit time is directly proportional to the total number of nuclei in the sample.
Statement $II$: The half-life of a radionuclide is the time required for the number of radioactive nuclei to reduce to half of its initial value at time $t = 0$.
In the light of the above statements, choose the most appropriate answer from the options given below:

$A$ radioactive material has a half-life of $8$ years. The activity of the material will decrease to about $1/8$ of its original value in .......... $years$.

If the half-life of a radioactive nucleus is $3$ days, nearly what fraction of the initial number of nuclei will decay on the third day? (Given, $\sqrt[3]{0.25} \approx 0.63$)

Consider a radioactive material of half-life $1.0 \, \text{minute}$. If one of the nuclei decays now,the next one will decay