Q49 / 100 · BITSAT 2024 (Memory Based Paper 2) · Mathematics

If $x$ is a complex root of the equation $\left|\begin{array}{ccc}1 & x & x \\ x & 1 & x \\ x & x & 1\end{array}\right|+\left|\begin{array}{ccc}1-x & 1 & 1 \\ 1 & 1-x & 1 \\ 1 & 1 & 1-x\end{array}\right|=0$, then $x^{2007}+x^{-2007}=$

  • A
    $1$
  • B
    $-1$
  • Correct answer
    C
    $-2$
  • D
    $2$

Solution

Expanding the two determinants, we get $\left(1-3 x^2+2 x^3\right)+\left(3 x^2-x^3\right)=0$ $\Rightarrow x^3+1=0$ $\Rightarrow x=-\omega,-\omega^2,-1$ $x^{2007}+x^{-2007}=-1-1=-2$

Hence, the correct answer is option: 'C'.

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