Complex Numbers Ques 107

If $\omega(\neq 1)$ be a cube root of unity and $\left(1+\omega^{2}\right)^{n}=\left(1+\omega^{4}\right)^{n}$, then the least positive value of $n$ is

(a) $2$

(b) $3$

(c) $5$

(d) $6$

(2004, 1M)

Show Answer

Answer:

Correct Answer: 107.(b)

Solution:

Formula:

Cube root of unity:

  1. Given, $\left(1+\omega^{2}\right)^{n}=\left(1+\omega^{4}\right)^{n}$

$\Rightarrow$ $(-\omega)^{n}$ $=\left(-\omega^{2}\right)^{n}$

$\left[\because \omega^{3}=1\right.$ and $\left.1+\omega+\omega^{2}=0\right]$

$\Rightarrow$ $\omega^{n}$ $=1$

$\Rightarrow$ $n$ $=3$ is the least positive value of $n$.



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