The activity of a radioactive sample is measured as $N_0$ counts per minute at $t = 0$ and $N_0/e$ counts per minute at $t = 5\, \text{minutes}$. The time (in minutes) at which the activity reduces to half of its initial value is

Which of the following statements are true regarding radioactivity?
$(I)$ All radioactive elements decay exponentially with time.
$(II)$ Half-life time of a radioactive element is the time required for one-half of the radioactive atoms to disintegrate.
$(III)$ The age of the Earth can be determined with the help of radioactive dating.
$(IV)$ Half-life time of a radioactive element is $50\%$ of its average life period.
Select the correct answer using the codes given below:

${ }_{92}^{238} U$ is known to undergo radioactive decay to form ${ }_{82}^{206} Pb$ by emitting alpha and beta particles. $A$ rock initially contained $68 \times 10^{-6} \text{ g}$ of ${ }_{92}^{238} U$. If the number of alpha particles that it would emit during its radioactive decay of ${ }_{92}^{238} U$ to ${ }_{82}^{206} Pb$ in three half-lives is $Z \times 10^{18}$,then what is the value of $Z$?

The rate of disintegration of a radioactive sample is $R$ and the number of atoms present at any time $t$ is $N$. When $\frac{R}{N}$ is taken along the $Y$-axis and $t$ is taken along the $X$-axis,the correct graph is:

The decay constant for the nuclide $^{64}Cu$ is $1.6 \times 10^{-5} \, s^{-1}$. Find the activity of a $1 \, mg$ sample of $^{64}Cu$ in $Ci$. (Given: Atomic mass of Copper = $64 \, g/mol$,$1 \, Ci = 3.7 \times 10^{10} \, Bq$)

The half-life of a radioactive sample is $T$. The fraction of the initial mass of the sample that decays in an interval $T / 2$ is