The half-life of a radioactive isotope $x$ is $50$ years. It decays to another element $y$ which is stable. The two elements $x$ and $y$ were found to be in the ratio of $1 : 7$ in a sample of a given rock. The age of the rock was estimated to be ........... $years$.

The half-life of a radioactive substance is $30 \text{ minutes}$. The time taken between $40 \%$ decay and $85 \%$ decay of the same radioactive substance 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?

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

Two samples $A$ and $B$ contain equal amount of radioactive substances. If $(1/8)^{th}$ of sample $A$ and $(1/128)^{th}$ of sample $B$ remain after $9 \text{ hours}$, then the ratio of half-life period of $B$ to that of $A$ is