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If $Z$ is a complex number such that $Z=-\bar{Z}$, then
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Verified Answer
The correct answer is:
$Z$ is purely imaginary.
Let $Z=x+i y$
Then, $\bar{Z}=x-i y$
Now, it is given that
$$
\begin{aligned}
& Z &=-\bar{Z} \\
\Rightarrow & x+i y &=-(x-i y) \\
\Rightarrow & x+i y &=-x+i y \\
\Rightarrow & & 2 x &=0 \\
\Rightarrow & x &=0
\end{aligned}
$$
$\therefore Z$ is purely imaginary.
Then, $\bar{Z}=x-i y$
Now, it is given that
$$
\begin{aligned}
& Z &=-\bar{Z} \\
\Rightarrow & x+i y &=-(x-i y) \\
\Rightarrow & x+i y &=-x+i y \\
\Rightarrow & & 2 x &=0 \\
\Rightarrow & x &=0
\end{aligned}
$$
$\therefore Z$ is purely imaginary.
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