<p>Let \(|z|=1\). Which of the following may be FALSE?</p>
Step-by-Step Solution
Key Concept: For |z|=1: 1/z=z̄, so z+1/z=z+z̄=2Re(z) — always real (B is TRUE not false). |z-1|\leq|z|+1=2 (C is TRUE). z \cdot z̄=|z|^2=1 (D TRUE). arg is in (-\pi,\pi] not all reals (A can be false as arg is bounded).
<p>(A): Principal argument $\in(-\pi,\pi]$, not all reals — A can be false. (B): $z+\bar{z}=2\text{Re}(z)$ is always real — B is always TRUE. (C): Triangle inequality gives $|z-1|\leq 2$ — TRUE. (D): $z\bar{z}=|z|^2=1$ — TRUE. So B and D are always true (never false), while A may be false depending on interpretation.</p>
Correct Answer: BD