If the system of equations has infinitely many solutions, then is equal to: [2025]

Question

If the system of equations

$2 x + λ y + 3 z = 5$

$3 x + 2 y - z = 7$

$4 x + 5 y + μ z = 9$

has infinitely many solutions, then $(λ^{2} + μ^{2})$ is equal to: [2025]

Smart Achievers · Accepted Answer

(3)

For infinitely man solutions, we have $△ = △_{1} = △_{2} = △_{3} = 0$

Now, $△ =$ $| \begin{matrix} 2 & λ & 3 \\ 3 & 2 & - 1 \\ 4 & 5 & μ \end{matrix} |$ $= 0$

$\Rightarrow 2 (2 μ + 5) - λ (3 μ + 4) + 3 (15 - 8) = 0$

$\Rightarrow 4 μ + 10 - 3 λ μ - 4 λ + 21 = 0$

$\Rightarrow 4 μ - 4 λ - 3 λ μ + 31 = 0$ ... (i)

Now, $△_{2} =$ $| \begin{matrix} 2 & 5 & 3 \\ 3 & 7 & - 1 \\ 4 & 9 & μ \end{matrix} |$ $= 0$

$\Rightarrow 2 (7 μ + 9) - 5 (3 μ + 4) + 3 (27 - 28)$

$\Rightarrow 14 μ + 18 - 15 μ - 20 - 3 = 0$

$\Rightarrow - μ + 18 - 23 = 0 \Rightarrow μ = - 5$

Substituting the value of $μ$ in (i), we get

$- 20 - 4 λ + 15 λ + 31 = 0$

$\Rightarrow 11 λ + 11 = 0 \Rightarrow λ = - 1$

$∴ λ^{2} + μ^{2} = {(- 1)}^{2} + {(- 5)}^{2} = 26$ .

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