TS EAMCET · Maths · Complex Number
If \(n, \mathrm{~K} \in \mathrm{~N}\) such that \(n \neq 3 \mathrm{~K}\), then \((\sqrt{3}+i)^{2 n}+(\sqrt{3}-i)^{2 n}=\)
- A \((-1)^n 2^{2 n+1}\)
- B \((-1)^{n+1} 2^{2 n+1}\)
- C \((-1)^{n+1} 2^{2 n}\)
- D \((-1)^{n+1} 2^n\)
Answer & Solution
Correct Answer
(C) \((-1)^{n+1} 2^{2 n}\)
Step-by-step Solution
Detailed explanation
\(\sqrt{3}+i = 2e^{i\frac{\pi}{6}}\) \(\sqrt{3}-i = 2e^{-i\frac{\pi}{6}}\) \((\sqrt{3}+i)^{2n}+(\sqrt{3}-i)^{2n} = (2e^{i\frac{\pi}{6}})^{2n} + (2e^{-i\frac{\pi}{6}})^{2n}\) \(= 2^{2n}e^{i\frac{n\pi}{3}} + 2^{2n}e^{-i\frac{n\pi}{3}}\)…
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