Some nuclei of a radioactive material are undergoing radioactive decay. The time gap between the instances when a quarter of the nuclei have decayed and when half of the nuclei have decayed is given as:

(where $\lambda$ is the decay constant)

A radioactive sample decays $\frac{7}{4}$ times its original quantity in $15$ minutes. The half-life of the sample is $……min$

The activity $R$ of an unknown radioactive nuclide is measured at hourly intervals. The results found are tabulated as follows:

$(i)$ Plot the graph of $R$ versus $t$ and calculate half-life from the graph.

$(ii)$ Plot the graph of $\ln \left( {\frac{R}{{{R_0}}}} \right) \to t$ versus $t$ and obtain the value of half-life from the graph.

Radioactive material $'A'$ has decay constant $8 \lambda$ and material $'B'$ has decay constant $ ' \lambda '$. Initially they have same number of nuclei . After what time, the ratio of number of nuclei of material $'B'$ to that $'A'$ will be $\frac{1}{e}$ ?

The half life of radioactive Radon is $3.8\, days$. The time at the end of which $1/20^{th}$ of the Radon sample will remain undecayed is  ………… $days$ (Given $log_{10}e = 0.4343$ )