(-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

If both mean and variance of $50$ observations $x_1, x_2, \ldots . ., x_{50}$ are equal to $16$ and $256$ respectively, then mean of $\left(x_1-5\right)^2,\left(x_2-5\right)^2, \ldots,\left(x_{50}-5\right)^2$ is

The domain of the function $f (x)=\log _{3+x}\left(x^2-1\right)$ is

If $\text{cosec x}+\cot \text{x}=\frac{1}{2},0<\text{x}<\frac{\pi}{2},$ then $\cos\text{x}$ is equal to

If A(0, 0), B(1, 3) and C(k, 0) are vertices of triangle ABC whose area is 3 sq.units, then the value of k is

The co-ordinates of the foci of the ellipse $3 x^2+4 y^2-12 x-8 y+4=0$, are

If $\sin 2 \theta=\frac{3}{4}$, then $\sin ^3 \theta+\cos ^3 \theta=$

Three numbers are chosen from 1 to 30. The probability that they are not consecutive, is

If $x=3+ i$, then $x^3-3 x^2-8 x+15=$

$\cos \frac{2 \pi}{15} \cos \frac{4 \pi}{15} \cos \frac{8 \pi}{15} \cos \frac{16 \pi}{15}=$

For the function $f (x)=\left\{\begin{array}{ll}\frac{ e ^{1 / x}-1}{ e ^{1 / x}+1}, & , x \neq 0 \\ 1 & , x=0\end{array}\right.$, which of the following is correct?