$A$ radioactive sample consists of two distinct species having an equal number of $N_0$ atoms initially. The mean-life of one species is $\tau$ and of the other is $5\tau$. The decay products in both cases are stable. The total number of radioactive nuclei at $t = 5\tau$ is

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.

If a radioactive substance reduces to $\frac{1}{16}$ of its original mass in $40$ days,what is its half-life in days?

The half-life of a radioactive sample is $5 \,s$. If the initial mass of the sample is $60 \,g$, then the time required to reduce the sample to $7.5 \,g$ is (in $\,s$)

The half-life of ${}_{92}^{238}U$ against $\alpha$-decay is $13.86 \times 10^{16} \,s$. The activity of a $1 \,g$ sample of ${}_{92}^{238}U$ is:

After $1$ hour,$1/16$ of the radioactive substance remains. Its half-life is ....... minutes.