If α,β are non real numbers satisfying x3−1=0 then the value

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Question

If

α, β

are non real numbers satisfying

x^{3} - 1 = 0

then the value of

∣ ∣∣ ∣ \begin{matrix} λ + 1 & α & β α & λ + β & 1 β & 1 & λ + α \end{matrix} ∣ ∣∣ ∣

is equal to

Options:

A . 0

B . λ3

C . λ3+1

D . λ3−1

Answer: Option B
:
B

x^{3} - 1 = 0 ∴ x = 1, ω, ω^{2}

Here,

α = ω, β = ω^{2} ∣ ∣∣∣ ∣ \begin{matrix} λ + 1 & ω & ω^{2} ω & λ + ω^{2} & 1 ω^{2} & 1 & λ + ω \end{matrix} ∣ ∣∣∣ ∣

Applying

C_{1} \to C_{1} + C_{2} + C_{3}

, then

∣ ∣∣∣ ∣ \begin{matrix} λ & ω & ω^{2} λ & λ + ω^{2} & 1 λ & 1 & λ + ω \end{matrix} ∣ ∣∣∣ ∣

Applying

R - 2 \to R_{2} - R_{1}

and

R_{3} \to R_{3} - R_{1}

, then we get

∣ ∣∣∣ ∣ \begin{matrix} λ & ω & ω^{2} 0 & λ + ω^{2} - ω & 1 - ω^{2} 0 & 1 - ω & λ + ω - ω^{2} \end{matrix} ∣ ∣∣∣ ∣ = λ ((λ + ω^{2} - ω) (λ + ω - ω^{2}) - (1 - ω) (1 - ω^{2})) = λ (λ^{2}) = λ^{3}

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