Write the definition of half life of radioactive substance and obtain its relation to decay  constant.

The half-life of a radioactive substance is $40$ years. How long will it take to reduce to one fourth of its original amount and what is the value of decay constant

The half life of a radioactive substance is $20$ minutes. In $……..\,minutes$ time,the activity of substance drops to $\left(\frac{1}{16}\right)^{ th }$ of its initial value.

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

How much mass of uranium to be destroyed per minute to operate a nuclear reactor of $600\,MW$