The nuclear activity of a radioactive element becomes $\left(\frac{1}{8}\right)^{\text{th}}$ of its initial value in $30\, \text{years}$. The half-life of the radioactive element is $....\, \text{years}$.

$A$ radioactive sample consists of two distinct species having an equal number of atoms initially. The mean life time of one species is $\tau$ and that of the other is $5 \tau$. The decay products in both cases are stable. $A$ plot is made of the total number of radioactive nuclei as a function of time. Which of the following figures best represents the form of this plot?

$A$ sample initially contains $10^{20}$ radioactive atoms. The number of $\alpha$-particles emitted in the third year is $0.3$ times the number of $\alpha$-particles emitted in the second year. How many $\alpha$-particles are emitted in the first year?

An element $X$ with a half-life of $1.4 \times 10^9$ years decays to form another stable element $Y$. $A$ sample is taken from a rock that contains both $X$ and $Y$ in the ratio $1:7$. If at the time of formation of the rock,$Y$ was not present in the sample,then the age of the rock in years is

The count rate of $10\,g$ of radioactive material was measured at different times and this has been shown in the figure. The half-life of the material and the total counts (approximately) in the first half-life period,respectively,are:

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: