If (1 + x + x²)ⁿ = a₀ + a₁x + a₂x² + ………+a_2nx²ⁿ, then the value of the determinant is always

The correct answer is: zero

(1 + x + x²)ⁿ = a₀ + a₁x + a₂x² + ………..+a_2nx²ⁿ …..(1)
Replacing x by, we get
(x² + x + 1)ⁿ = a₀x²ⁿ + a₁x^{2n –1} + a₂x^{2n –2} + …….+a²ⁿ …….(2)
Equating the coefficients of like terms from (1) and (2), we get
a₀ = a_2n, a₁ = a_{2n –1},……………a_r = a_{2n –r} Þ a_{n –1} = a_{n + 1}, a_{n –3} = a_{n +3}, a_{n + 7} = a_{n –7}
So value of determinant is zero as second and third columns are identical.