At $t = 0$,the number of active nuclei in a sample is $N_0$. How many nuclei will decay in the time interval between its first mean life and second half-life?

$A$ radioactive material is reduced to $1/8$ of its original amount in $3$ days. If $8 \times 10^{-3} \, kg$ of the material is left after $5$ days,the initial amount of the material is $....... \, g$.

In a sample of radioactive material,what fraction of the initial number of active nuclei will remain undisintegrated after half of a half-life of the sample?

The half-life of a radioactive substance is $100 \ s$. How many grams of the radioactive substance will remain after $5 \ min$ if the initial amount is $8 \ g$ (in $g$)?

$A$ $280\, \text{day}$ old radioactive substance shows an activity of $6000\, \text{dps}$. $140\, \text{days}$ later, its activity becomes $3000\, \text{dps}$. What was its initial activity? ......... $\text{dps}$

Following statements related to radioactivity are given below:
$(A)$ Radioactivity is a random and spontaneous process and is dependent on physical and chemical conditions.
$(B)$ The number of undecayed nuclei in the radioactive sample decays exponentially with time.
$(C)$ Slope of the graph of $\log_{e}$ (no. of undecayed nuclei) $Vs.$ time represents the negative reciprocal of mean life time $(-\frac{1}{\tau})$.
$(D)$ Product of decay constant $(\lambda)$ and half-life time $(T_{1/2})$ is constant,equal to $\ln(2)$.
Choose the most appropriate answer from the options given below.