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Question

# If the value of determinant ∣∣ ∣∣x+1αβαx+β1β1x+α∣∣ ∣∣is equal to -8, then the value of x, is (where α, β are non real cube roots of unity)

A

±2

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B

-2

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C

0

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D

1

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Solution

## The correct option is B -2 We have α=ω and β=ω2△=∣∣ ∣∣xαβxx+β1x1x+α∣∣ ∣∣ [C1→C1+C2+C3 and using 1+ω+ω2=0]=∣∣ ∣∣xαβ0x+β−α1−β01−αx+α−β∣∣ ∣∣ [R2→R2−R1 and R3→R3−R1]=x[x2−(α−β)2−(1−α)(1−β)]=x(x2−α2−β2+2αβ−1+α+β−αβ)=x[x2−ω2−ω+2−1+ω+ω2−1] [using ω2=1]=x3 According to the given condition, we have x3=−8 gives x=−2

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