Question

Let $Z$ and $W$ be complex numbers such that $|Z| = |W| $ 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 $\overline{W}=\pi.$

• A. Statement 1 is true, Statement 2 is false
• B. Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1
• C. Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1
• D. Statement 1 is false, Statement 2 is true

Answer 1

Answer: (A)

Let $\left|Z\right|=\left|W\right|=r$
$\Rightarrow Z=re^{i\theta}, W=re^{i\phi}$
where $\theta+\phi=\pi$
$\therefore \,=-\overline{W}=re^{-i\phi}$
Now, $Z=re^{i\left(\pi-\phi\right)}=re^{i\pi}\times e^{-i\phi}$
$=-\overline{W}$
Thus, statement-1 is true but statement-2 is false.