Complex Numbers 5 Question 6
6. 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)
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Answer:
Correct Answer: 6. (b)
Solution:
- 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 $n$. |
