For α, β∈ℝ, suppose the system of linear equations x−y+z=52x+2y+αz=83x−y+4z=βhas infinitely many solutions. Then α and β are the roots of

1−1122α3−14=0⇒α=4[AD]=1−11522483−14β=1−115021−1021β−15⇒β−15=−1⇒β = 14∴α, β are roots of x2 - 18x + 56 = 0

For α, β∈ℝ, suppose the system of linear equations x−y+z=52x+2y+αz=83x−y+4z=βhas infinitely many solutions. Then α and β are the roots of

1−1122α3−14=0⇒α=4[AD]=1−11522483−14β=1−115021−1021β−15⇒β−15=−1⇒β = 14∴α, β are roots of x2 - 18x + 56 = 0

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For $α, β \in ℝ$ , suppose the system of linear equations
$\begin{array}{l} x - y + z = 5 \\ 2 x + 2 y + αz = 8 \\ 3 x - y + 4 z = β \end{array}$
has infinitely many solutions. Then $α$ and $β$ are the roots of

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x² + 18x + 56 = 0

x² - 18x + 56 = 0

x² + 14x + 24 = 0

x² - 10x + 16 = 0

answer is A.

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Detailed Solution

$\begin{array}{l} |\begin{array}{ccc} 1 & - 1 & 1 \\ 2 & 2 & α \\ 3 & - 1 & 4 \end{array}| = 0 \Rightarrow α = 4 \\ [AD] = [\begin{array}{cccc} 1 & - 1 & 1 & 5 \\ 2 & 2 & 4 & 8 \\ 3 & - 1 & 4 & β \end{array}] = [\begin{array}{cccc} 1 & - 1 & 1 & 5 \\ 0 & 2 & 1 & - 1 \\ 0 & 2 & 1 & β - 15 \end{array}] \Rightarrow β - 15 = - 1 \end{array}$
$\Rightarrow β$ = 14
$∴ α, β$ are roots of x² - 18x + 56 = 0