TS EAMCET · Maths · Complex Number
If \(z=\frac{-1-i \sqrt{3}}{2}\), then \(\sum_{k=1}^{2022}\left(z^k+\frac{1}{z^k}\right)^2=\)
- A 0
- B 2022
- C 4044
- D 1011
Answer & Solution
Correct Answer
(C) 4044
Step-by-step Solution
Detailed explanation
\(z = \omega^2\) and \(\frac{1}{z} = \omega\), where \(\omega=e^{i2\pi/3}\) is a cube root of unity. \(z^k+\frac{1}{z^k} = \omega^{2k}+\omega^k\). If \(k \equiv 0 \pmod 3\), \((\omega^{2k}+\omega^k)^2 = (1+1)^2 = 4\). If \(k \equiv 1 \pmod 3\),…
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