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  4. The least positive integer n for which ( (1 + i √3/1 - i √3) )n = 1 , is :

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Q. The least positive integer n for which $\left( \frac{1 + i \sqrt{3}}{1 - i \sqrt{3}} \right)^n = 1 $, is :

JEE MainJEE Main 2018Complex Numbers and Quadratic Equations

Solution:

$\begin{pmatrix}\frac{1+i\sqrt{3}}{1-i\sqrt{3}}\end{pmatrix}^{n} = 1$
$\begin{pmatrix}\frac{-2\omega^{2}}{-2\omega}\end{pmatrix}^{n} = 1$
$\omega^{n} = 1$
least positive integer value of n is 3.