Let the system of equations x + 2 y + 3 z = 5 , 2 x + 3 y + z = 9 , 4 x + 3 y + z = have infinite number of solutions. Then is equal to: [2024]

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Solution

Q.
Let the system of equations $x + 2 y + 3 z = 5,$ $2 x + 3 y + z = 9,$ $4 x + 3 y + λ z = μ$ have infinite number of solutions. Then $λ + 2 μ$ is equal to: [2024]

1 22

2 17

3 15

4 28

Ans.
(2)

Given system of equations can be written as, $A X = B$

Where $A = [\begin{matrix} 1 & 2 & 3 \\ 2 & 3 & 1 \\ 4 & 3 & λ \end{matrix}], X = [\begin{matrix} x \\ y \\ z \end{matrix}],$ and $B = [\begin{matrix} 5 \\ 9 \\ μ \end{matrix}]$

For infinitely many solutions, $D = 0 \Rightarrow$ $| \begin{matrix} 1 & 2 & 3 \\ 2 & 3 & 1 \\ 4 & 3 & λ \end{matrix} |$ $= 0$

$\Rightarrow λ = - 13$

Also, $D_{1} = 0 \Rightarrow$ $| \begin{matrix} 5 & 2 & 3 \\ 9 & 3 & 1 \\ μ & 3 & - 13 \end{matrix} |$ $= 0 \Rightarrow μ = 15$

So, $λ + 2 μ = - 13 + 30 = 17$

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