The half-life period of a radioactive sample is $T$. If a fraction $x$ disintegrates in time $t$,what fraction will decay in time $\frac{t}{2}$?

Carbon-$14$ decays with a half-life of about $5,800$ years. In a sample of bone,the ratio of Carbon-$14$ to Carbon-$12$ is found to be $\frac{1}{4}$ of what it is in free air. This bone may belong to a period about $x$ centuries ago,where $x$ is nearest to:

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 half-life of a radioactive substance is $20 \, min$. The approximate time interval $(t_2 - t_1)$ between the time $t_2$ when $\frac{2}{3}$ of it has decayed and the time $t_1$ when $\frac{1}{3}$ of it has decayed is .......... $min$.

Two radioactive materials $R_1$ and $R_2$ have decay constants $6 \lambda$ and $\lambda$, respectively. The half-life of $R_2$ is $1.4 \times 10^{17} \,s$. Initially, they contain the same number of nuclei. The time at which the ratio of the remaining nuclei of $R_2$ to that of $R_1$ will be $e$ is (Let $\ln 2 = 0.7$):

Draw a graph showing the variation of decay rate with the number of active nuclei.