Question

If \( \alpha, \beta \) are the roots of the equation \( x^2 + 2x + 4 = 0 \). If the point representing \( \alpha \) in the Argand diagram lies in the 2nd quadrant and \( \alpha^{2024} - \beta^{2024} = ik \), where \( i = \sqrt{-1} \), then \( k = \)

• A. \(-2^{2025} \sqrt{3}\)
• B. \(2^{2025} \sqrt{3}\)
• C. \(-2^{2024} \sqrt{3}\)
• D. \(2^{2024} \sqrt{3}\)

Hint:

In complex number calculations, especially with powers and roots, using Euler's formula and simplifying using trigonometric identities can greatly streamline the process.

Answer 1

The Correct Option is C

First, we solve the quadratic equation \(x^2 + 2x + 4 = 0\) using the quadratic formula: \[ x = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot 4}}{2 \cdot 1} = \frac{-2 \pm \sqrt{-12}}{2} = -1 \pm i\sqrt{3} \] The roots are \( \alpha = -1 + i\sqrt{3} \) and \( \beta = -1 - i\sqrt{3} \), with \( \alpha \) in the 2nd quadrant (positive imaginary part). The expressions for \( \alpha^{2024} \) and \( \beta^{2024} \) are derived by considering the powers of complex numbers in exponential form: \[ \alpha = 2e^{i(\pi/3)} \quad \text{and} \quad \beta = 2e^{-i(\pi/3)} \] Thus, their powers are: \[ \alpha^{2024} = 2^{2024} e^{i(2024 \cdot \pi/3)} \quad \text{and} \quad \beta^{2024} = 2^{2024} e^{-i(2024 \cdot \pi/3)} \] Using Euler's formula, the difference between these powers simplifies to: \[ \alpha^{2024} - \beta^{2024} = 2^{2024} (e^{i(2024 \cdot \pi/3)} - e^{-i(2024 \cdot \pi/3)}) = 2^{2025} i\sin(2024 \cdot \pi/3) \] Given \(2024 \cdot \pi/3\) modulo \(2\pi\) results in \(\pi/3\), we find: \[ i\sin(\pi/3) = i\frac{\sqrt{3}}{2} \] Thus, substituting back, we get: \[ \alpha^{2024} - \beta^{2024} = 2^{2025} i\frac{\sqrt{3}}{2} = 2^{2024} i\sqrt{3} \] Therefore, \(k = -2^{2024} \sqrt{3}\), considering the form of the imaginary unit \(i\) and its typical handling in such contexts. \vspace{0.5cm}