What is the relationship between the decay constant $\lambda$ and the half-life $\tau_{1/2}$?

Consider a radioactive material of half-life $1.0 \, \text{minute}$. If one of the nuclei decays now,the next one will decay

The activity of a radioactive sample is measured as $N_0$ counts per minute at $t = 0$ and $N_0/e$ counts per minute at $t = 5\, \text{minutes}$. The time (in minutes) at which the activity reduces to half of its initial value is

The half-life of a radioactive substance is $48$ hours. How much time will it take to disintegrate to its $\frac{1}{16}$th part?

$99\%$ of the nuclei of a radioactive element decay between which of the following times?

At a given instant,say $t = 0$,two radioactive substances $A$ and $B$ have equal activities. The ratio $\frac{R_B}{R_A}$ of their activities after time $t$ decays with time $t$ as $e^{-3t}$. If the half-life of $A$ is $\ln 2$,the half-life of $B$ is: