If the system of equations x + 2y –3z = 2 has infinitely many solutions, then is equal to : [2025]

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Solution

Q.
If the system of equations

x + 2y –3z = 2

$2 x + λ y + 5 z = 5$

$14 x + 3 y + μ z = 33$

has infinitely many solutions, then $λ + μ$ is equal to : [2025]

1 13

2 11

3 12

4 10

Ans.
(3)

Given : x + 2y – 3z = 2

$2 x + λ y + 5 z = 5$

$14 x + 3 y + μ z = 33$

For infinitely many solutions, we have $∆ = 0$

$\Rightarrow$ $| \begin{matrix} 1 & 2 & - 3 \\ 2 & λ & 5 \\ 14 & 3 & μ \end{matrix} |$ $= 0$

$\Rightarrow 1 (λ μ - 15) - 2 (2 μ - 70) - 3 (6 - 14 λ) = 0$

$\Rightarrow λ μ + 42 λ - 4 μ + 107 = 0$

We also have, $△_{1} = 2 λ μ + 99 λ - 10 μ + 225$

$△_{2} = μ - 13, △_{3} = 5 λ + 5$

Now, $△_{2} = 0 and △_{3} = 0$

$∴ 13 - μ = 0 and 5 λ + 5 = 0 \Rightarrow μ = 13 \Rightarrow λ = - 1$

Thus, we get $λ + μ = 13 + (- 1) = 13 - 1 = 12$

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