The system of equations x = y + z = 6 x + 2y + 5x = 9 has no solution if [2025]

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Solution

Q.
The system of equations

x + y + z = 6

x + 2y + 5x = 9

$x + 5 y + λ z = μ$

has no solution if [2025]

1 $λ = 15, μ \neq 17$

2 $λ \neq 17, μ \neq 18$

3 $λ = 17, μ = 18$

4 $λ = 17, μ \neq 18$

Ans.
(4)

Let $A = [\begin{matrix} 1 & 1 & 1 \\ 1 & 2 & 5 \\ 1 & 5 & λ \end{matrix}] ∴$ $| A |$ $=$ $| \begin{matrix} 1 & 1 & 1 \\ 1 & 2 & 5 \\ 1 & 5 & λ \end{matrix} |$ $= 0$

$\Rightarrow 1 (2 λ - 25) - (λ - 5) + 1 (5 - 2) = 0 \Rightarrow λ = 17$

$D_{z} =$ $| \begin{matrix} 1 & 1 & 6 \\ 1 & 2 & 9 \\ 1 & 5 & μ \end{matrix} |$ $\neq 0$ $[\begin{matrix} ∵ If | A | = 0 then atleast one D_{x}, D_{y}, D_{z} \\ is non zero for inconsistent system \end{matrix}]$

$\Rightarrow μ \neq 18$ .

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