The number of values of α for which the system of equations $\begin{array}{l} x + y + z = α, αx + 2 αy + 3 z = - 1, x + 3 αy + 5 z = 4 \end{array}$ is inconsistent, is

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answer is B.

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Detailed Solution

$\begin{array}{l} x + y + z = α \\ αx + 2 αy + 3 z = - 1 \\ x + 3 αy + 5 z = 4 \end{array}$
Has inconsistent solution
$\begin{array}{l} D = |\begin{array}{ccc} 1 & 1 & 1 \\ α & 2 α & 3 \\ 1 & 3 α & 5 \end{array}| = 0 \\ \Rightarrow 1 (10 α - 9 α) - 1 (5 α - 3) + 1 (3 α^{2} - 2 α) = 0 \Rightarrow 3 α^{2} - 6 α + 3 = 0 \\ \Rightarrow α^{2} - 2 α + 1 = 0 \\ \Rightarrow (α - 1)^{2} = 0 \Rightarrow α = 1 \end{array}$
For $α = 1$
$\begin{array}{l} D_{1} = |\begin{array}{ccc} 1 & 1 & 1 \\ - 1 & 2 & 3 \\ 4 & 3 & 5 \end{array}| \\ = (10 - 9) - (- 5 - 12) + (- 3 - 8) \\ = 1 + 17 - 11 \neq 0 \end{array}$
For α = 1 the system of equation has Inconsistent solution