If \(\alpha, \beta\) are the roots of \(x^2-2 x+4=0\), for \(n \in \mathbf{N}\), what is the value of…
If \(\alpha, \beta\) are the roots of \(x^2-2 x+4=0\), for \(n \in \mathbf{N}\), what is the value of \(\alpha^n+\beta^n=\)
- \(2^{n+2} \cos \left(\frac{n \pi}{3}\right)\)
- \(2^{n+1} \cos \left(\frac{n \pi}{3}\right)\)
- \(2^{n+1} \cos \left(\frac{n \pi}{6}\right)\)
- \(2^{n+2} \cos \left(\frac{n \pi}{6}\right)\)
Solution
Given,
\(\alpha, \beta\) are roots of \(x^2-2 x+4=0\)
\(\begin{aligned}
& x=\frac{2 \pm \sqrt{4-4 \cdot 1 \cdot 4}}{2 \cdot 1} \\
& x=\frac{2 \pm \sqrt{-12}}{2} \\
& x=1 \pm \sqrt{3 i} \\
& x=1+\sqrt{3 i}(o r) 1-\sqrt{3 i} \\
& x=2\left(\frac{1}{2}+\frac{\sqrt{3}}{2} i\right)=2\left(\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}\right) \\
& \alpha=2 \cdot e^{\frac{i \pi}{3}}
\end{aligned}\)
Similarly, \(\beta=2 \cdot e^{\frac{-i \pi}{3}}\)
\(\begin{aligned}
\alpha^n+\beta^n & =\left(2 e^{\frac{i \pi}{3}}\right)^n+\left(2 e^{\frac{-i \pi}{3}}\right)^n=2^n\left(e^{\frac{i n \pi}{3}}+e^{\frac{-i n \pi}{3}}\right) \\
& =2^n\left(\cos \frac{n \pi}{3}+i \sin \frac{n \pi}{3}+\cos \frac{n \pi}{3}-i \sin \frac{n \pi}{3}\right) \\
& =2^n 2 \cos \frac{n \pi}{3}=2^{n+1} \cdot \cos \frac{n \pi}{3}
\end{aligned}\)
Hence, option (b) is correct.
Asked in: AP EAMCET 2020 (18 Sep Shift 2)
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