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JEE Main · 2024 · 27 Jan 2024 · Shift 1 · Complex Numbers

If

α

satisfies the equation

x^{2} + x + 1 = 0

and

(1 + α)^{7} = A + B α + C α^{2}

,

A, B, C \geq 0

, then

5 (3 A - 2 B - C)

is equal to ____.

Answer: 5

Solution

x^{2} + x + 1 = 0

has roots

ω, ω^{2}

with

ω^{3} = 1

and

1 + ω + ω^{2} = 0

. Take

α = ω

, so

1 + α = 1 + ω = - ω^{2}

.

(1 + α)^{7} = (- ω^{2})^{7} = - ω^{14} = - ω^{2} = 1 + ω = 1 + α

(using

14 \equiv 2 (mod 3)

and

- ω^{2} = 1 + ω

). Comparing with

A + B α + C α^{2}

:

A = 1, B = 1, C = 0

.

5 (3 A - 2 B - C) = 5 (3 - 2 - 0) = 5

JEE Main · 2024 · 27 Jan 2024 · Shift 1 · Integer answer

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