Let If the system of equations 3 x + 5 y + z = 3 7 x + 11 y - 9 z = 2 97 x + 155 y - 189 z = has infinitely many solutions, then is equal to: [2024]

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Solution

Q.
Let $λ, μ \in R .$ If the system of equations

$3 x + 5 y + λ z = 3$

$7 x + 11 y - 9 z = 2$

$97 x + 155 y - 189 z = μ$

has infinitely many solutions, then $μ + 2 λ$ is equal to: [2024]

1 24

2 27

3 25

4 22

Ans.
(3)

$3 x + 5 y + λ z = 3$

$7 x + 11 y - 9 z = 2$

$97 x + 155 y - 189 z = μ$

$[\begin{matrix} 3 & 5 & λ \\ 7 & 11 & - 9 \\ 97 & 155 & - 189 \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} 3 \\ 2 \\ μ \end{matrix}]$

$A X = B$

For infinitely many solutions,

$| A |$ $= 0 and (adj A) B = 0$

$\Rightarrow$ $| \begin{matrix} 3 & 5 & λ \\ 7 & 11 & - 9 \\ 97 & 155 & - 189 \end{matrix} |$ $= 0$

$\Rightarrow - 2052 + 2250 + 18 λ = 0$

$\Rightarrow λ = - 11$

$adj A = [\begin{matrix} - 684 & - 760 & 76 \\ 450 & 500 & - 50 \\ 18 & 20 & - 2 \end{matrix}]$

$(adj A) B = [\begin{matrix} - 684 & - 760 & 76 \\ 450 & 500 & - 50 \\ 18 & 20 & - 2 \end{matrix}] [\begin{matrix} 3 \\ 2 \\ μ \end{matrix}] = [\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}]$

$\Rightarrow 54 + 40 - 2 μ = 0$

$\Rightarrow 2 μ = 94 \Rightarrow μ = 47$

Hence, $μ + 2 λ = 25$

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