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The equation $a x + a^{2} y + z = 0, b x + b^{2} y + z = 0$ and $c x + c^{2} y + z = 0$ have only one solution (0,0,0). The coefficients a, b, c are in G.P then the common ratio of the G.P cannot be equal to

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$0, \pm 1$

none of these

$4, \pm 5$

$2, \pm 3$

answer is A.

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Detailed Solution

$Δ = |\begin{matrix} a & a^{2} & 1 \\ b & b^{2} & 1 \\ c & c^{2} & 1 \end{matrix}| \neq 0$

$\Rightarrow (a - b) (b - c) (c - a) \neq 0$

$\Rightarrow a^{3} r {(1 - r)}^{3} (1 + r) \neq 0$

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