If x is a complex root of the equation | beginmatrix1&x&x x&1&x x&x&1 endmatrix|+| beginmatrix1-x&1&1 1&1-x&1 1&1&1-x endmatrix|=0 , then x2007+ x-2007=

Question

If x is a complex root of the equation | beginmatrix1&x&x x&1&x x&x&1 endmatrix|+| beginmatrix1-x&1&1 1&1-x&1 1&1&1-x endmatrix|=0 , then x2007+ x-2007= (A) 1 (B) -1 (C) -2 (D) 2

Tardigrade Admin · Accepted Answer

Expanding the two determinants, we get (1-3x2+2x3)+(3x2-x3)=0 ⇒ x3+1=0 ⇒ x=-ω, -ω2, -1 x2007+x-2007=-1-1=-2 .

Q. If $x$ is a complex root of the equation
$∣ ∣ 1 x x x 1 x x x 1 ∣ ∣ + ∣ ∣ 1 - x 11 1 1 - x 1 11 1 - x ∣ ∣ = 0$ , then $x^{2007} + x^{- 2007} =$

$1$

$- 1$

$- 2$

$2$

Expanding the two determinants, we get

$(1 - 3 x^{2} + 2 x^{3}) + (3 x^{2} - x^{3}) = 0$

$\Rightarrow x^{3} + 1 = 0 \Rightarrow x = - ω, - ω^{2}, - 1$

$x^{2007} + x^{- 2007} = - 1 - 1 = - 2$ .

Q. If x is a complex root of the equation ∣∣​1xx​x1x​xx1​∣∣​+∣∣​1−x11​11−x1​111−x​∣∣​=0 , then x2007+x−2007=

1

−1

−2

2

Expanding the two determinants, we get (1−3x2+2x3)+(3x2−x3)=0 ⇒x3+1=0⇒x=−ω,−ω2,−1 x2007+x−2007=−1−1=−2 .

Q. If $x$ is a complex root of the equation
$∣ ∣ 1 x x x 1 x x x 1 ∣ ∣ + ∣ ∣ 1 - x 11 1 1 - x 1 11 1 - x ∣ ∣ = 0$ , then $x^{2007} + x^{- 2007} =$

$1$

$- 1$

$- 2$

$2$

Expanding the two determinants, we get

$(1 - 3 x^{2} + 2 x^{3}) + (3 x^{2} - x^{3}) = 0$

$\Rightarrow x^{3} + 1 = 0 \Rightarrow x = - ω, - ω^{2}, - 1$

$x^{2007} + x^{- 2007} = - 1 - 1 = - 2$ .