AP EAMCET · Maths · Quadratic Equation
If \(\alpha, \beta\) are the roots of the equation \(x^2-2 x+4=0\) and for any \(n \in \mathrm{N}, \alpha^n+\beta^n=k \cos \frac{n \pi}{3}\) then \(k=\)
- A \(2^n\)
- B \(2^{n+1}\)
- C \(2^n-1\)
- D \(2^n+1\)
Answer & Solution
Correct Answer
(B) \(2^{n+1}\)
Step-by-step Solution
Detailed explanation
\(x = \frac{2 \pm \sqrt{(-2)^2 - 4(1)(4)}}{2} = \frac{2 \pm \sqrt{-12}}{2} = 1 \pm i\sqrt{3}\) \(\alpha = 1 + i\sqrt{3} = 2 \left(\cos \frac{\pi}{3} + i \sin \frac{\pi}{3}\right)\)…
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