Question

Number of complex numbers z satisfying $z^3=\overline{z}$ is

• A. 1
• B. 2
• C. 4
• D. 5

Answer 1

The correct option is D 5

$z^3=\overline{z}$

let $z=x+iy$

then, $(x+iy{)}^3=x-iy$

$(x^3-3x{y}^2)+(3x^{2}y-y^3)i=x-iy$

$x^3-3x{y}^2=x$ $3x^{2}y-y^3=-y$

$x(x^2-3y^2-1)=0$ and $y(3x^2-y^2+1)=0$

Here, we get all value of $(x,y)$

i.e, $(0,0)(\frac{i}{\sqrt{2}},\frac{i}{\sqrt{2}}),(\frac{-i}{\sqrt{2}},\frac{i}{\sqrt{2}}),(\frac{i}{\sqrt{2}},\frac{-i}{\sqrt{2}})$

and $(\frac{-i}{\sqrt{2}},\frac{-i}{\sqrt{2}})$

total 5 values