Q112 / 180 · COMEDK 2019 · Mathematics

If $Z$ is a complex number such that $Z=-\bar{Z}$, then

  • A
    $Z$ is purely real.
  • Correct answer
    B
    $Z$ is purely imaginary.
  • C
    $Z$ is any complex number.
  • D
    Real part of $Z$ is the same as its imaginary part.

Solution

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.

Hence, the correct answer is option: 'B'.

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