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Let $Z$ and $W$ be complex numbers such that $\left| Z \right| = \left| W \right|,$ and arg $Z$ denotes the principal argument of $Z.$
Statement $1:$ If arg $Z+$ arg $W = \pi ,$ then $Z = -\overline W $.
Statement $2:$ $\left| Z \right| = \left| W \right|,$ implies arg $Z-$ arg $\overline W = \pi .$
Statement $1$ is true, Statement $2$ is false.
Statement $1$ is true, Statement $2$ is true, Statement $2$ is a correct explanation for Statement $1$
Statement $1$ is true, Statement $2$ is true, Statement $2$ is not a correct explanation for Statement $1.$
Statement $1$ is false, Statement $2$ is true
Solution
Let $|Z|=|W|=r$
$\Rightarrow Z=r e^{i \theta}, W=r e^{i \phi}$
where $\theta+\phi=\pi$
$\therefore \bar{W}=r e^{-i \phi}$
$\mathrm{Now}, Z=r e^{i(\pi-\phi)}=r e^{i \pi} \times e^{-i \phi}=-r e^{-i \phi}$
$=-\vec{W}$
Thus, statement $-\,1$ is true but statement $-\,2$ is false
