The decay constant for a radioactive nuclide is $1.5 \times 10^{-5} \, s^{-1}$. The molar mass of the substance is $60 \, g \, mol^{-1}$,$(N_A = 6 \times 10^{23})$. The activity of $1.0 \, \mu g$ of the substance is $....... \times 10^{10} \, Bq$.

The activity of a radioactive sample is $6000 \, dps$ after $280$ days. After another $140$ days,the activity decreases to $3000 \, dps$. The initial activity of the radioactive sample (in $dps$) is ........

At $t = 0$,the number of active nuclei in a sample is $N_0$. How many nuclei will decay in the time interval between its first mean life and second half-life?

Half-lives of two radioactive substances $A$ and $B$ are,respectively,$20 \ min$ and $40 \ min$. Initially,the samples of $A$ and $B$ have an equal number of nuclei. After $80 \ min$,the ratio of the remaining number of $A$ and $B$ nuclei is

$A$ sample of radioactive material $A$,which has an activity of $10\, mCi$ $(1\, Ci = 3.7 \times 10^{10}\, \text{decays/s})$,has twice the number of nuclei as another sample of different radioactive material $B$,which has an activity of $20\, mCi$. The correct choices for half-lives of $A$ and $B$ would then be respectively:

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: