If alpha, beta are the roots of the equation | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) The given equation is $x^2 - 2\sqrt{3}x + 4 = 0$.Using the quadratic formula,the roots are $\alpha, \beta = \frac{2\sqrt{3} \pm \sqrt{12 - 16}}{2}

Vedclass · Accepted Answer

$2^{2025}$

Vedclass · Answer

(B) The given equation is $x^2 - 2\sqrt{3}x + 4 = 0$.Using the quadratic formula,the roots are $\alpha, \beta = \frac{2\sqrt{3} \pm \sqrt{12 - 16}}{2} = \sqrt{3} \pm i$.In polar form,$\alpha = 2(\cos \frac{\pi}{6} + i \sin \frac{\pi}{6})$ and $\beta = 2(\cos \frac{-\pi}{6} + i \sin \frac{-\pi}{6})$.Then $\alpha^{2024} = 2^{2024}(\cos \frac{2024\pi}{6} + i \sin \frac{2024\pi}{6})$ and $\beta^{2024} = 2^{2024}(\cos \frac{-2024\pi}{6} + i \sin \frac{-2024\pi}{6})$.Note that $\frac{2024\pi}{6} = \frac{1012\pi}{3} = 337\pi + \frac{\pi}{3}$.$\alpha^{2024} - \beta^{2024} = 2^{2024} [2i \sin(\frac{2024\pi}{6})] = 2^{2024} [2i \sin(337\pi + \frac{\pi}{3})] = 2^{2024} [2i (-\sin \frac{\pi}{3})] = 2^{2024} [2i (-\frac{\sqrt{3}}{2})] = -i \sqrt{3} \cdot 2^{2024}$.Therefore,$\frac{2}{\sqrt{3}} |\alpha^{2024} - \beta^{2024}| = \frac{2}{\sqrt{3}} | -i \sqrt{3} \cdot 2^{2024} | = \frac{2}{\sqrt{3}} \cdot \sqrt{3} \cdot 2^{2024} = 2^{2025}$.

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

Similar Questions

If $1, \omega, \omega^2$ are the three cube roots of unity,then the value of $(a + b\omega + c\omega^2)^3 + (a + b\omega^2 + c\omega)^3$ is equal to,given that $a + b + c = 0$.

$\sum_{k=1}^6 \left[ \sin \frac{2 k \pi}{7} - i \cos \frac{2 k \pi}{7} \right]$ is equal to

If $x_n = \cos \frac{\pi}{2^n} + i \sin \frac{\pi}{2^n}$,then $\prod_{n=1}^{\infty} x_n$ is equal to

The number of distinct solutions of the equation $x^{11}-x^7+x^4-1=0$ is

If $\alpha, \beta \in \mathbb{C}$ are distinct roots of the equation $x^2 - x + 1 = 0$,then $\alpha^{101} + \beta^{107}$ is equal to:

Vedclass Test Series

Exam Paper Generator

Online Exam Module