Q107 / 180 · KCET 2008 · Mathematics

If is a complex number such that $\mathrm{z}=-\overline{\mathrm{z}}$, then

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

Solution

Let $\mathrm{z}=\mathrm{x}+\mathrm{iy}$ Given, $\quad \mathrm{z}=-\overline{\mathrm{z}}$ $$ \begin{array}{lc} \therefore & x+i y=-(\overline{x+i y}) \\ \Rightarrow & x+i y=-(x-i y) \\ \Rightarrow & 2 x=0 \\ & x=0 \end{array} $$ Hence, $\mathrm{z}$ is a purely imaginary.

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

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