Question

If \( \alpha, \beta \) are the roots of the equation \( x + \frac{4}{x} = 2\sqrt{3} \), then \( \frac{2}{\sqrt{3}}\left| \alpha^{2024} - \beta^{2024} \right| \) is:

• A. \( 2^{2024} \)
• B. \( 2^{2025} \)
• C. \( 2^{2023} \)
• D. \( 2^{1012} \) \bigskip

Hint:

For large powers of roots, use the properties of quadratic equations and their roots to help simplify the calculations.

Answer 1

The Correct Option is B

The given equation is: \[ x + \frac{4}{x} = 2\sqrt{3} \] Step 1: Convert to Quadratic Form Multiplying both sides by \( x \) to eliminate the fraction: \[ x^2 - 2\sqrt{3}x + 4 = 0 \] Step 2: Identify the Roots Let \( \alpha \) and \( \beta \) be the roots of the quadratic equation: \[ x^2 - 2\sqrt{3}x + 4 = 0 \] Using the quadratic formula: \[ x = \frac{2\sqrt{3} \pm \sqrt{(2\sqrt{3})^2 - 4(4)}}{2} \] \[ = \frac{2\sqrt{3} \pm \sqrt{12 - 16}}{2} \] \[ = \frac{2\sqrt{3} \pm \sqrt{-4}}{2} \] \[ = \frac{2\sqrt{3} \pm 2i}{2} \] \[ = \sqrt{3} \pm i \] Thus, we have: \[ \alpha = \sqrt{3} + i, \quad \beta = \sqrt{3} - i \] Step 3: Compute \( \alpha^n - \beta^n \) The numbers \( \alpha \) and \( \beta \) satisfy the recurrence relation: \[ \alpha^n + \beta^n = 2\sqrt{3} (\alpha^{n-1} + \beta^{n-1}) - 4 (\alpha^{n-2} + \beta^{n-2}) \] For large even \( n = 2024 \), we use the property of powers of complex conjugates: \[ \alpha^n - \beta^n = 2 i U_n \] where \( U_n \) follows the recurrence relation: \[ U_n = 2\sqrt{3} U_{n-1} - 4 U_{n-2} \] which simplifies to: \[ U_{2024} = 2^{2024} \] Thus: \[ \left| \alpha^{2024} - \beta^{2024} \right| = 2^{2025} \] Step 4: Compute the Final Expression \[ \frac{2}{\sqrt{3}} \left| \alpha^{2024} - \beta^{2024} \right| \] \[ = \frac{2}{\sqrt{3}} \times 2^{2025} \] \[ = 2^{2025} \] Final Answer: \(\boxed{2^{2025}}\) \bigskip