If α , β ∈ C are the distinct roots of the equation x 2 - x + 1 = 0 , then α 101 + β 107 is equal to

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Question: Answered & Verified by Expert

If

α, β \in C

are the distinct roots of the equation

x^{2} - x + 1 = 0,

then

α^{101} + β^{107}

is equal to

MathematicsComplex NumberJEE MainJEE Main 2018 (08 Apr)

Options:

A $2$
B $- 1$
C $0$
D $1$

Solution:

2861 Upvotes Verified Answer

The correct answer is:

1

$x^{2} - x + 1 = 0$

Roots are $- ω, - ω^{2}$ .

$α^{101} + β^{107} = {(- ω)}^{101} + {(- ω^{2})}^{107}$

$= - ω^{2} - ω (∵ ω^{3} = 1)$

$= - (ω^{2} + ω)$

$= 1 (∵ 1 + ω + ω^{2} = 0)$

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