Question

If $z = \cos\left(\frac{\pi}{3} \right) - i \sin \left(\frac{\pi }{3}\right),$ the $z^{2} - z +1 $ is equal to

• A. 0
• B. 1
• C. -1
• D. $\frac{\pi}{2}$

Answer 1

The Correct Option is A

The correct option is(A): 0.

We have,
\(z =\cos \frac{\pi}{3}-i \sin \frac{\pi}{3}\)
\(=\frac{1}{2}-\frac{i \sqrt{3}}{2}\)
\(=\frac{1-\sqrt{3} i}{2}\)
\(=-\left[\frac{-1+\sqrt{3} i}{2}\right]\)
\(=-w \left[\because w=\frac{-1+\sqrt{3} i}{2}\right]\)
Now, \(z^{2}-z+1=(-w)^{2}-(-w)+1\)
\(=w^{2}+w+1\)
\(=0 \,\,\left[\because 1+w+w^{2}=0\right]\)