Multiple choice

For the equation $\begin{vmatrix} 1 & x & { x }^{ 2 } \ { x }^{ 2 } & 1 & x \ x & { x }^{ 2 } & 1 \end{vmatrix}=0$

  1. There are exactly two distinct roots

  2. There is one pair of equation real roots

  3. There are three pairs of equal roots

  4. Modulus of each root is 2
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C Correct answer
Explanation

The determinant of the matrix is -(x^3-1)^2 = 0, which implies x^3 = 1. The roots are 1, omega, and omega^2, where omega is a complex cube root of unity. Each root appears twice, leading to three pairs of equal roots.

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