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 percentage of ${}^{235}U$ presently on Earth is $0.72\%$ and the rest $(99.28\%)$ may be taken to be ${}^{238}U$. Assume that all uranium on Earth was produced in a supernova explosion long ago with the initial ratio ${}^{235}U / {}^{238}U = 2.0$. How long ago did the supernova event occur? (Take the half-lives of ${}^{235}U$ and ${}^{238}U$ to be $7.1 \times 10^8$ years and $4.5 \times 10^9$ years respectively).

The graph in the figure shows how the count-rate $A$ of a radioactive source as measured by a Geiger counter varies with time $t$. The relationship between $A$ and $t$ is (Assume $\ln 12 = 2.6$):

$A$ radioactive nucleus can decay by two different processes. The half-lives of the first and second decay processes are $5 \times 10^3$ years and $10^5$ years respectively. Then, the effective half-life of the nucleus is

From $N$ atoms of a radioactive element,$n$ alpha particles are emitted per second. The half-life of the element is ......

$A$ radioactive substance decays by $10\%$ in $5$ days. What percentage of the original substance will remain after $20$ days (in $\%$)?