Question

if z= cos(π/3)-i sin (π/3) ,then z^2 -z+1 is equal to

Answer 1

$z = \cos( \frac{\pi}{3} ) - i \sin( \frac{\pi}{3} ) \\ z = (\frac{1}{2} - i \frac{ \sqrt{3} }{2} ) \\ z = \frac{1 - i \sqrt{3} }{2}$
${z}^{2} = \frac{1}{4} \times (1 - 3 - i2 \sqrt{3} ) \\ = \frac{1}{4} \times - 2(1 + i \sqrt{3} ) \\ \frac{ - 1}{2} (1 + i \sqrt{3} )$
now put the values in equation
$= \frac{ - 1}{2} (1 + i \sqrt{3}) - \frac{1}{2} (1 - i \sqrt{3} ) + 1$
=
$= - \frac{1}{2} - \frac{1}{2} + 1 \\ = - 1 + 1 \\ = 0$
option A. zero is the answer