Question 12

X, Y, Z coplanar means the coefficient matrix (in terms of x, y, z, which are linearly independent) is singular: Matrix M = [[α, β, -1], [-1, α, β], [β, -1, α]] (circulant). det = α³ + β³ + (-1)³ − 3·α·β·(-1) = α³ + β³ − 1 + 3αβ. Using identity a³+b³+c³−3abc = (a+b+c)(a²+b²+c²−ab−bc−ca) with a=α, b=β, c=−1: det = (α+β−1)(α²+β²+1−αβ+α+β). Second factor = ½[(α−β)² + (α+1)² + (β+1)²] > 0 for distinct α, β. So α+β−1 = 0, i.e., α+β = 1, hence α+β−3 = −2. Check x, y, z independent: det [[1,2,3],[2,3,1],[3,1,2]] = 1(6−1) −2(4−3) +3(2−9) = 5 −2 −21 = −18 ≠ 0. Good.

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