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:
- 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 $n$.
BETA
