If ω is an imaginary cube root of unity, then (1 + ω - ω^2)^7 is equal to

If ω is an imaginary cube root of unity, then (1 + ω -… — JEE Main 1998 Mathematics PYQ

1998

\omega

is an imaginary cube root of unity, then

(1 + \omega - \omega^2)^7

is equal to

A.
$128\omega$
B.
$-128\omega$
C.
$128\omega^2$
D.
$-128\omega^2$

Show answer and solution

Correct option: D

Solution

Since ω is a primitive cube root of unity, 1 + ω + ω² = 0, so 1 + ω = -ω².

Therefore (1 + ω - ω²) = -ω² - ω² = -2ω².

Raising to the 7th power: (-2ω²)⁷ = -128 · ω¹⁴ = -128 · ω² (since ω³ = 1, so ω¹⁴ = ω²).

The answer is -128ω².

[1998]

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