At time $t = 0$,the mass of a radioactive element sample is $10 \, g$. After a time interval equal to two mean lives,what mass of the sample will approximately remain in $g$?

The nuclide $^{133}I$ is radioactive,with a half-life of $8.04 \, days$. At noon on $January \, 1$,the activity of a certain sample is $600 \, Bq$. The activity at noon on $January \, 24$ will be

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:

The half-life of $Ra^{226}$ is $1620$ years. Calculate the number of atoms that decay in one second in $1 \ g$ of radium (Avogadro number $= 6.023 \times 10^{23}$).

$A$ sample of living carbon-containing matter has an activity of $15$ decays/minute per gram of carbon-$14$. $A$ sample from the Indus Valley civilization has an activity of $9$ decays/minute per gram of carbon-$14$. Given the half-life of $C^{14}$ is $5730$ years, find the age of the Indus Valley civilization.

The radioactive sources $A$ and $B$ have half-lives of $2 \ hr$ and $4 \ hr$ respectively,and initially contain the same number of radioactive atoms. At the end of $2 \ hr$,their rates of disintegration are in the ratio: