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Question

Mathematics Question on Complex Numbers and Quadratic Equations

If z=cos⁡(π3)−isin⁡(π3),z = \cos\left(\frac{\pi}{3} \right) - i \sin \left(\frac{\pi }{3}\right), the z2−z+1z^{2} - z +1 is equal to

A

0

B

1

C

-1

D

π2\frac{\pi}{2}

Answer

0

Explanation

Solution

The correct option is(A): 0.

We have,
z=cos⁡π3−isin⁡π3z =\cos \frac{\pi}{3}-i \sin \frac{\pi}{3}
=12−i32=\frac{1}{2}-\frac{i \sqrt{3}}{2}
=1−3i2=\frac{1-\sqrt{3} i}{2}
=−[−1+3i2]=-\left[\frac{-1+\sqrt{3} i}{2}\right]
=−w[∵w=−1+3i2]=-w \left[\because w=\frac{-1+\sqrt{3} i}{2}\right]
Now, z2−z+1=(−w)2−(−w)+1z^{2}-z+1=(-w)^{2}-(-w)+1
=w2+w+1=w^{2}+w+1
=0  [∵1+w+w2=0]=0 \,\,\left[\because 1+w+w^{2}=0\right]