WBJEE · Maths · Complex Number
Let \(\alpha\) and \(\beta\) be the roots of \(x^{2}+x+1=0 .\) If \(n\) be a positive integer, then \(\alpha^{n}+\beta^{n}\) is
- A \(2 \cos \frac{2 n \pi}{3}\)
- B \(2 \sin \frac{2 n \pi}{3}\)
- C \(2 \cos \frac{n \pi}{3}\)
- D \(2 \sin \frac{n \pi}{3}\)
Answer & Solution
Correct Answer
(A) \(2 \cos \frac{2 n \pi}{3}\)
Step-by-step Solution
Detailed explanation
We have, \(x^{2}+x+1=0\) \(\Rightarrow \quad x=\frac{-1 \pm \sqrt{3} i}{2}\) \(\Rightarrow \quad \alpha=\frac{-1+\sqrt{3} i}{2}\) and \(\beta=\frac{-1-\sqrt{3} i}{2}\) or \(\quad \alpha=e^{\frac{i2\pi}{3}}\) and \(\beta=e^{\frac{-2\pi i}{3}}\) \(\therefore\)…
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