Let S be the set of values of, for which the system of equations 6 x - 3 y + 3 z = 42 , 2 x + 6 y + 4 z = 1 , 3 x + 2 y + 3 z = has no solution. Then 12 is equal to ___________ . [2023]

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Solution

Q.
Let S be the set of values of $λ$ , for which the system of equations

$6 λ x - 3 y + 3 z = 4 λ^{2},$

$2 x + 6 λ y + 4 z = 1,$

$3 x + 2 y + 3 λ z = λ$ has no solution. Then $12 \sum_{λ \in S} | λ |$ is equal to ___________ . [2023]

Ans.
(24)

For no solution, $Δ = 0$

$\Rightarrow$ $| \begin{matrix} 6 λ & - 3 & 3 \\ 2 & 6 λ & 4 \\ 3 & 2 & 3 λ \end{matrix} |$ $= 0$

$\Rightarrow 6 λ (18 λ^{2} - 8) + 3 (6 λ - 12) + 3 (4 - 18 λ) = 0$

$\Rightarrow 18 λ^{3} - 14 λ - 4 = 0 \Rightarrow λ = 1, \frac{- 1}{3}, \frac{- 2}{3}$

$For each λ, Δ_{1} =$ $| \begin{matrix} 6 λ & - 3 & 4 λ^{2} \\ 2 & 6 λ & 1 \\ 3 & 2 & λ \end{matrix} |$ $\neq 0$

$Then, 12 \sum_{λ \in S} | λ | = 12 (1 + \frac{1}{3} + \frac{2}{3}) = 24$

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