An ordered pair - - for which the system of linear equations 1- alpha x- beta y-z-2 - lalpha x-1- beta y-z-3 alpha x- beta y-2 z-21 has a unique solution- is-

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Linear System of Equations

An ordered pa...

Question

An ordered pair $(α, β)$ for which the system of linear equations
$(1 + α) x + β y + z = 2 α x + (1 + β) y + z = 3 α x + β y + 2 z = 2$
has a unique solution, is:

(1, - 3)

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(- 3, 1)

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(- 4, 2)

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(2, 4)

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Solution

The correct option is D $(2, 4)$
The system of linear equation are
$(1 + α) x + β y + z = 2 α x + (1 + β) y + z = 3 α x + β y + 2 z = 2$
for unique solution, $Δ \neq 0$
$\Rightarrow ∣ ∣ ∣ ∣ \begin{matrix} 1 + α & β & 1 α & 1 + β & 1 α & β & 2 \end{matrix} ∣ ∣ ∣ ∣ \neq 0$
Apply $R_{1} \to R_{1} - R_{2}$
$\Rightarrow ∣ ∣ ∣ ∣ \begin{matrix} 1 & - 1 & 0 α & 1 + β & 1 α & β & 2 \end{matrix} ∣ ∣ ∣ ∣ \neq 0$
$\Rightarrow 1 (2 + 2 β - β) + 1 (2 α - α) + 0 \neq 0 \Rightarrow α + β + 2 \neq 0 \Rightarrow α + β \neq - 2$
Hence, $(2, 4)$ will satisfy this condition.

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An ordered pair (α,β) for which the system of linear equations (1+α)x+βy+z=2αx+(1+β)y+z=3αx+βy+2z=2 has a unique solution, is:

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