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If the system of equations $2 x + y - z = 5, 2 x - 5 y + λ z = μ, x + 2 y - 5 z = 7$ has infinitely many solutions then ${(λ + μ)}^{2} + {(λ - μ)}^{2}$ is equal to

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916

904

920

912

answer is A.

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Detailed Solution

$Δ = 0$

$|\begin{matrix} 2 & 1 & - 1 \\ 2 & - 5 & λ \\ 1 & 2 & - 5 \end{matrix}| = 0$

$λ = 17$

$Δ_{1} = 0$

$|\begin{matrix} 5 & 1 & - 1 \\ μ & - 5 & 17 \\ 7 & 2 & - 5 \end{matrix}| = 0$

$μ = - 13$

$= {(λ + μ)}^{2} + {(λ - μ)}^{2}$

$= {(4)}^{2} + {(30)}^{2}$

$= 16 + 900 = 916$

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