Half-lives of two radioactive nuclei $A$ and $B$ are $10 \, minutes$ and $20 \, minutes$,respectively. If,initially,a sample has an equal number of nuclei,then after $60 \, minutes$,the ratio of the number of decayed nuclei of $A$ and $B$ will be:

An ancient discovery found a sample where $75 \%$ of the original carbon $(C^{14})$ remains. The age of the sample is: $\left(T_{1/2}(C^{14}) = 5730 \text{ years}, \ln 0.5 = -0.7, \ln 0.75 = -0.3\right)$ (in $\text{ years}$)

Two radioactive materials $Y_1$ and $Y_2$ initially contain the same number of nuclei. Their decay constants are $9 \lambda \ s^{-1}$ and $6 \lambda \ s^{-1}$ respectively. The time after which the ratio of the number of undecayed nuclei of $Y_1$ and $Y_2$ becomes $\frac{1}{e}$ is:

The activity of a radioactive element decreases to one-third of its original activity $R_0$ in $9$ years. After a further $9$ years,its activity will be

The radioactivity of a given sample of whisky due to tritium (half-life $12.5 \text{ years}$) was found to be only $3\%$ of that measured in a recently purchased bottle marked "$7 \text{ years}$ old". The sample must have been prepared about:

$A$ radioactive nucleus decays by two different processes. The half-life of the first process is $5$ minutes and that of the second process is $30\,s$. The effective half-life of the nucleus is calculated to be $\frac{\alpha}{11}\,s$. The value of $\alpha$ is $..............$