The half-life of a radioactive substance is $T$. The time taken for disintegrating $\frac{7}{8}$th part of its original mass will be

The activity of a radioactive sample decreases to $(1/3)$ of its original value in $3\, \text{days}$. Then, in $9\, \text{days}$, its activity will become:

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:

The fraction of the initial number of radioactive nuclei which remain undecayed after half of a half-life of the radioactive sample is

The initial activity of a radioactive element is $1600$. After $t = 8 \, s$,the activity becomes $100$. What is the activity at $t = 6 \, s$?

$A$ radioactive material has an initial amount $16 \, gm$. After $120 \, days$ it reduces to $1 \, gm$. The half-life of the radioactive material is .......... $days$.