The relation between half-life $(T)$ and decay constant $(\lambda)$ is

The activity of a radioactive sample decreases to $\left(\frac{1}{3}\right)$ of its original value in $3 \ days$. Then,in $9 \ days$,its activity reduces to:

Given below are two statements:
Statement $I$: The law of radioactive decay states that the number of nuclei undergoing the decay per unit time is directly proportional to the total number of nuclei in the sample.
Statement $II$: The half-life of a radionuclide is the time required for the number of radioactive nuclei to reduce to half of its initial value at time $t = 0$.
In the light of the above statements, choose the most appropriate answer from the options given below:

The rate of disintegration was observed to be $10^{17}$ disintegrations per second when its half-life period is $1445$ years. The original number of particles is:

The decay constant of a radioactive element is $1.5 \times 10^{-9} \text{ s}^{-1}$. Its mean life in seconds will be:

$A$ certain radioactive substance reduces to $25\%$ of its initial value in $16\ days$. Its half-life is .......... $days$.