System of linear equations

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Question

The absolute difference of values of $α^{10}$ for which the system of equations $x + y + z = 1,$ $x + 2 y + 4 z = α, x + 4 y + 10 z = α^{2}$ is consistent is,

Moderate

Solution

$(2) - (1) \Rightarrow y + 3 z = α - 1$
$(3) - (1) \Rightarrow 3 y + 9 z = α^{2} - 1$
$\Rightarrow 3 (y + 3 z) = a^{2} - 1$
$∴$ $\Rightarrow 3 (α - 1) = α^{2} - 1$
$\Rightarrow α^{2} - 3 α + 2 = 0 \Rightarrow α = 1, 2$
$∴$ Absolute difference $= 2^{10} - 1^{10} = 1024 - 1 = 1023$

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