Complex Numbers 5 Question 9
10. If $\omega$ is an imaginary cube root of unity, then $\left(1+\omega-\omega^{2}\right)^{7}$ is equal to
(a) $128 w$
(b) $-128 \omega$
(c) $128 \omega^{2}$
(d) $-128 \omega^{2}$
(1998, 2M)
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Answer:
Correct Answer: 10. (d)
Solution:
- $\left(1+\omega-\omega^{2}\right)^{7}=\left(-\omega^{2}-\omega^{2}\right)^{7}$
$\left[\because 1+\omega+\omega^{2}=0\right]$
$$ =\left(-2 \omega^{2}\right)^{7}=(-2)^{7} \omega^{14}=-128 \omega^{2} $$
