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 un-decayed 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 reciprocal of mean life time $(\tau)$.

$(D)$ Product of decay constant ( $\lambda$ ) and half-life time $\left(T_{1 / 2}\right)$ is not constant.

Choose the most appropriate answer from the options given below

Certain radio-active substance reduces to $25\%$ of its value in $16$ days. Its half-life is ........ $days$ 

The activity of a radioactive sample

The disintegration rate of a certain radioactive sample at any instant is $4250$ disintegrations per minute.$10$ minutes later, the rate becomes $2250$ disintegrations per minute. The approximate decay cons $.........\min^{-1}$

In a radioactive material, fraction of active material remaining after time $t$ is $\frac{9}{16}$ The fraction that was remaining after $\frac{t}{2}$ is 

In a radioactive decay chain reaction, ${ }_{90}^{230} Th$ nucleus decays into ${ }_{84}^{214} Po$ nucleus. The ratio of the number of $\alpha$ to number of $\beta^{-}$particles emitted in this process is. . . . .