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

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Solution

Q.
If the system of equations

$(λ - 1) x + (λ - 4) y + λ z = 5$

$λ x + (λ - 1) y + (λ - 4) z = 7$

$(λ + 1) x + (λ + 2) y - (λ + 2) z = 9$

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

1 20

2 12

3 6

4 10

Ans.
(2)

For infinitely many solutions,

$D =$ $| \begin{matrix} (λ - 1) & (λ - 4) & λ \\ λ & (λ - 1) & (λ - 4) \\ (λ + 1) & (λ + 2) & - (λ + 2) \end{matrix} |$ $= 0$ ... (i)

On solving (i) along column (1), we get

$2 λ^{2} - 5 λ - 3 = 0$

$\Rightarrow (2 λ + 1) (λ - 3) = 0$

$\Rightarrow λ = 3 or - \frac{1}{2}$ ... (ii)

$D_{2} = 0 \Rightarrow$ $| \begin{matrix} (λ - 1) & 5 & λ \\ λ & 7 & (λ - 4) \\ (λ + 1) & 9 & - (λ + 2) \end{matrix} |$ $= 0$

$\Rightarrow 2 λ^{2} - 13 λ + 21 = 0$

$\Rightarrow (2 λ - 7) (λ - 3) = 0$

$\Rightarrow λ = 7 / 2 or λ = 3$ ... (iii)

Thus, $λ = 3$ (From (ii) and (iii))

$∴ λ^{2} + λ = {(3)}^{2} + 3 = 9 + 3 = 12$ .

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