If left(frac{3}{2}+i frac{sqrt{3}}{2}right)^{ | vedclass

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

Question

(D) Let $z = \frac{3}{2} + i \frac{\sqrt{3}}{2} = \sqrt{3} \left( \frac{\sqrt{3}}{2} + i \frac{1}{2} \right) = \sqrt{3} \left( \cos \frac{\pi}{6} + i

Vedclass · Accepted Answer

$\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$

Vedclass · Answer

(D) Let $z = \frac{3}{2} + i \frac{\sqrt{3}}{2} = \sqrt{3} \left( \frac{\sqrt{3}}{2} + i \frac{1}{2} \right) = \sqrt{3} \left( \cos \frac{\pi}{6} + i \sin \frac{\pi}{6} \right) = \sqrt{3} e^{i \frac{\pi}{6}}$.Then, $\left( \frac{3}{2} + i \frac{\sqrt{3}}{2} \right)^{50} = (\sqrt{3})^{50} e^{i \frac{50\pi}{6}} = 3^{25} e^{i \frac{25\pi}{3}}$.Since $\frac{25\pi}{3} = 8\pi + \frac{\pi}{3}$, we have $e^{i \frac{25\pi}{3}} = e^{i \frac{\pi}{3}} = \cos \frac{\pi}{3} + i \sin \frac{\pi}{3} = \frac{1}{2} + i \frac{\sqrt{3}}{2}$.Thus, $3^{25} (x + iy) = 3^{25} \left( \frac{1}{2} + i \frac{\sqrt{3}}{2} \right)$.Comparing real and imaginary parts, we get $x = \frac{1}{2}$ and $y = \frac{\sqrt{3}}{2}$.

If $\left(\frac{3}{2}+i \frac{\sqrt{3}}{2}\right)^{50}=3^{25}(x+i y)$ where $x$ and $y$ are real, then the ordered pair $(x, y)$ is

Similar Questions

The value of $(\frac{-1 + i\sqrt{3}}{2})^{18} + (\frac{-1 - i\sqrt{3}}{2})^{18}$ is

If $\alpha$ and $\beta$ are the roots of the equation $x^2+x+1=0$, then the equation whose roots are $\alpha^{19}$ and $\beta^7$ is

If $z = \cos \theta + i \sin \theta$,then $z^r + (\bar{z})^r = $

If the cube roots of unity are $1, \omega, \omega^2$,then the roots of the equation $(x - 1)^3 + 8 = 0$ are

If $\omega$ is a cube root of unity,then the value of $(1 - \omega + \omega^2)^5 + (1 + \omega - \omega^2)^5$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module