(-1+ i 3)^ 20 is equal to - Vidyadip

MCQ

$(-1+ i \sqrt{3})^{20}$ is equal to

A
$2^{20}(-1+ i \sqrt{3})^{20}$
B
$2^{20}(1-i \sqrt{3})^4$
C
$2^{20}(-1-i \sqrt{3})^{20}$
✓
None of these

✓

Answer

Correct option: D.

None of these

(D)
Let $z =-1+ i \sqrt{3}, r =\sqrt{1+3}=2$
$\theta=\tan ^{-1}\left(\frac{\sqrt{3}}{-1}\right)=\frac{2 \pi}{3}$
$\therefore \quad z=2\left(\cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}\right)$
$\therefore \quad(z)^{20}=\left[2\left(\cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}\right)\right]^{20}$
$=2^{20}\left(\cos \frac{2 \pi}{3}+ i \sin \frac{2 \pi}{3}\right)^{20}$
$=2^{20}\left(-\frac{1}{2}+ i \frac{\sqrt{3}}{2}\right)^{20}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "MCQ" from Complex Numbers→Complex Numbers — all question types→Maths STD 11 Science question papers→Maharashtra Board STD 11 Science question papers→Maharashtra Board question paper generator→

Similar questions

The line parallel to the X-axis and passing through the intersection of the lines ax + 2by + 3b = 0 and bx - 2ay - 3a = 0 where $(a, b) \neq(0,0)$ is

The value of $\cos(36^\circ-\text{A})\cos(36^\circ+\text{A})+\cos(54^\circ+\text{A})\cos(54^\circ-\text{A})$ is :

$\cos ^2 \alpha+\cos ^2\left(\alpha+120^{\circ}\right)+\cos ^2\left(\alpha-120^{\circ}\right)$ is equal to

$\cos ^3 110^{\circ}+\cos ^3 10^{\circ}+\cos ^3 130^{\circ}=$

The function $f(x)=\frac{|x|}{x^2+2 x}, x \neq 0$ and $f(0)=0$ is not continuous at $x=0$ because

If $\text{y}=1+\frac{\text{x}}{1!}+\frac{\text{x}^2}{2!}+\frac{\text{x}^3}{3!}+\dots,$ then $\frac{\text{dy}}{\text{dx}}=$

The standard deviation of the data:

$x$	$1$	$a$	$a^2$	$....$	$a^n$
$f$	$^nC_0$	$^nC_1$	$^nC_2$	$....$	$^nC_2$

If in an infinite $G.P$., first term is equal to $10$ times the sum of all successive terms, the its common ratio is :

The equation of the ellipse whose foci are ( $\pm 5,0$ ) and one of its directrix is 5x = 36 is

The value of $\cos 20^{\circ} \cos 40^{\circ} \cos 60^{\circ} \cos 80^{\circ}$ is equal to