KCET · Maths · Complex Number
If is a complex number such that \(\mathrm{z}=-\overline{\mathrm{z}}\), then
- A is purely real
- B is purely imaginary
- C is any complex number
- D real part of is the same as its imaginary part
Answer & Solution
Correct Answer
(B) is purely imaginary
Step-by-step Solution
Detailed explanation
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.
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.
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