Q.If the number of nuclei in a radioactive sample at a given time is N, what will be the number at the end of two half-lives? (A) N/2 (B) N/4 (C) 3N/4 (D) N/8

[!TLDR] Each half-life halves the population; after two half-lives, $N\rightarrow N/ …

By definition, one half-life reduces the population of parent nuclei to exactly half its value at the start of that interval, regardless of how much time has already elapsed. Starting from N nuclei, after one half-life the number remaining is $N/2$; applying the SAME halving rule again over the next half-life (not starting over, but continuing from $N/2$) gives $\frac{1}{2}\times\frac{N}{2}=\frac{N}{4}$ remaining after two half-lives in total. This also follows direct …

Apply the halving rule twice in succession (or equivalently use $N=N_0(1/2 …

Assuming the population reaches zero after the second half-life, or incorrectly computing $N/2-N/2=0$, instead of recognising that each half-l …