$A$ radioactive substance has a half-life of $1$ year. The fraction of this material that would remain after $5$ years will be

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:

After $3$ hours,only $0.25 \,mg$ of a pure radioactive material is left. If the initial mass was $2 \,mg$,then the half-life of the substance is ...... $hr$.

$A$ radioactive sample of half-life $ 10 $ days contains $ 1000x $ nuclei. The number of original nuclei present after $ 5 $ days is: (in $x$)

The graph which represents the correct variation of logarithm of activity $(\log A)$ versus time $t$ is:

From $N$ atoms of a radioactive element,$n$ alpha particles are emitted per second. The half-life of the element is ......