If the system of equations 2 x + y - z = 5 , 2 x - 5 y + , x + 2 y - 5 z = 7 has infinitely many solutions, then 2 + 2 is equal to: [2023]v

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Solution

Q.
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: [2023]

1 912

2 916

3 904

4 920

Ans.
(2)

$2 x + y - z = 5$

$2 x - 5 y + λ z = μ$

$x + 2 y - 5 z = 7$

The system of equations has infinitely many solutions when $Δ = Δ_{1} = Δ_{2} = Δ_{3} = 0$

$Δ = 0$

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

$\Rightarrow 2 (25 - 2 λ) - 1 (- 10 - λ) - 1 (4 + 5) = 0$

$\Rightarrow 50 - 4 λ + 10 + λ - 9 = 0 \Rightarrow - 3 λ = - 51 \Rightarrow λ = 17$

$Δ_{3} = 0$

$| \begin{matrix} 2 & 1 & 5 \\ 2 & - 5 & μ \\ 1 & 2 & 7 \end{matrix} |$ $= 0$ $\Rightarrow 2 (- 35 - 2 μ) - 1 (14 - μ) + 5 (4 + 5) = 0$

$\Rightarrow - 70 - 4 μ - 14 + μ + 45 = 0 \Rightarrow - 3 μ = 39 \Rightarrow μ = - 13$

$∴$ ${(λ + μ)}^{2} + {(λ - μ)}^{2} = {(17 - 13)}^{2} + {(17 + 13)}^{2} = 4^{2} + 30^{2} = 16 + 900 = 916$

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