TS EAMCET · Maths · Complex Number
\(\alpha, \beta\) are the roots of the equation \(x^2+2 x+4=0\). If the point representing \(\alpha\) in the Argand diagram lies in the 2nd quadrant and \(\alpha^{2024}-\beta^{2024}=i k,(i=\sqrt{-1})\), then \(k=\)
- A \(-2^{2025} \sqrt{3}\)
- B \(2^{2025} \sqrt{3}\)
- C \(-2^{2024} \sqrt{3}\)
- D \(2^{2024} \sqrt{3}\)
Answer & Solution
Correct Answer
(C) \(-2^{2024} \sqrt{3}\)
Step-by-step Solution
Detailed explanation
\begin{aligned} & \text { } x^2+2 x+4=0 \Rightarrow(x+1)^2+3=0 \\ & x=-1 \pm \sqrt{3} i \Rightarrow \alpha=-1+\sqrt{3} i \quad\left(\alpha \text { lies in } 2^{\text {nd }} \text { quadrant }\right) \\ & \alpha=2 \operatorname{cis}\left(\frac{2 \pi}{3}\right) \Rightarrow…
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