If ( 1 + i 3 1 - i 3 )^n is an integer, then n is - Vidyadip

MCQ

If ${\left( {\frac{{1 + i\sqrt 3 }}{{1 - i\sqrt 3 }}} \right)^n}$ is an integer, then $n$ is

A
$1$
B
$2$
✓
$3$
D
$4$

✓

Answer

Correct option: C.

$3$

c
(c) $\frac{{1 + i\sqrt 3 }}{{1 - i\sqrt 3 }} = \left( {\frac{{1 + i\sqrt 3 }}{{1 - i\sqrt 3 }}} \right)\,\left( {\frac{{1 + i\sqrt 3 }}{{1 + i\sqrt 3 }}} \right) = \frac{{ - 2 + i2\sqrt 3 }}{4}$
$ = \,\frac{{ - 1 + i\sqrt 3 }}{2} = \omega $
${\left( {\frac{{1 + i\sqrt 3 }}{{1 - i\sqrt 3 }}} \right)^n} = {\omega ^n} = {\omega ^3} = 1 \Rightarrow n = 3$.

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

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

The minimum area of a triangle formed by any tangent to the ellipse $\frac{{{x^2}}}{{16}} + \frac{{{y^2}}}{{81}} = 1$ and the co-ordinate axes is

The solution of the differential equation, $x^2 \frac{{dy}}{{dx}} \cdot \cos \frac{1}{x} - y \sin \frac{1}{x} = - 1,$ where $y \rightarrow - 1$ as $x \rightarrow \infty$ is

Let $f:[0,2] \rightarrow R$ be the function defined by

$f ( x )=(3-\sin (2 \pi x )) \sin \left(\pi x -\frac{\pi}{4}\right)-\sin \left(3 \pi x +\frac{\pi}{4}\right)$

If $\alpha, \beta \in[0,2]$ are such that $\{x \in[0,2]: f(x) \geq 0\}=[\alpha, \beta]$, then the value of $\beta-\alpha$ is. . . . . . . . .

Sum of infinite number of terms in $G.P.$ is $20$ and sum of their square is $100$. The common ratio of $G.P.$ is

$\mathop {\lim }\limits_{x \to 4} \left[ {\frac{{{x^{3/2}} - 8}}{{x - 4}}} \right] = $

If a complex number $z$ statisfies the equation $x + \sqrt 2 \,\,\left| {z + 1} \right|\,+ \,i\, = \,0,$ then $\left| z \right|$ is equal to

The determinant $\left| {\,\begin{array}{*{20}{c}}a&b&{a - b}\\b&c&{b - c}\\2&1&0\end{array}\,} \right|$ is equal to zero if $a,b,c$ are in

The integral $80 \int_{0}^{\frac{\pi}{4}}\left(\frac{\sin \theta+\cos \theta}{9+16 \sin 2 \theta}\right) d \theta$ is equal to :

If ${\tan ^{ - 1}}2x + {\tan ^{ - 1}}3x = \frac{\pi }{4}$, then $x =$

A tangent to the hyperbola $\frac{{{x^2}}}{4} - \frac{{{y^2}}}{2} = 1$ meets $x-$ axis at $P$ and $y-$ axis at $Q$. Lines $PR$ and $QR$ are drawn such that $OPRQ$ is a rectangle (where $O$ is the origin). Then $R$ lies on