If the system of equations 11 x + y + z = - 5 2 x + 3 y + 5 z = 3 8 x - 19 y - 39 z = has infinitely many solutions, then is equal to: [2024]

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Solution

Q.
If the system of equations

$11 x + y + λ z = - 5$

$2 x + 3 y + 5 z = 3$

$8 x - 19 y - 39 z = μ$

has infinitely many solutions, then $λ^{4} - μ$ is equal to: [2024]

1 49

2 51

3 47

4 45

Ans.
(3)

We have, $11 x + y + λ z = - 5$

$2 x + 3 y + 5 z = 3$

$8 x - 19 y - 39 z = μ$

has infinitely many solutions

$∴$ $Δ =$ $| \begin{matrix} 11 & 1 & λ \\ 2 & 3 & 5 \\ 8 & - 19 & - 39 \end{matrix} |$ $= 0$

$\Rightarrow 11 (- 22) - 1 (- 118) + λ (- 62) = 0$

$\Rightarrow - 124 - 62 λ = 0$

$\Rightarrow λ = - 2$

Also, $Δ_{3} =$ $| \begin{matrix} 11 & 1 & - 5 \\ 2 & 3 & 3 \\ 8 & - 19 & μ \end{matrix} |$ $= 0$

$\Rightarrow 11 (3 μ + 57) - 1 (2 μ - 24) - 5 (- 38 - 24)$

$\Rightarrow 31 μ + 961 = 0 \Rightarrow μ = - 31$

So, $λ^{4} - μ = {(- 2)}^{4} + 31 = 16 + 31 = 47$

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