QJEEM23Ma788f: JEE Main 2023 Mathematics - Class 12 JEEM23Ma788f Solution: For , suppose the system of linear equations has infinitely…

Question

Verified Solution • 2025–26 Session moderate

For

\alpha, \beta \in \mathbb{R}

, suppose the system of linear equations

\begin{aligned}&amp; x-y+z=5 \\&amp; 2 x+2 y+\alpha z=8 \\&amp; 3 x-y+4 z=\beta\end{aligned}

has infinitely many solutions. Then

\alpha

and

\beta

are the roots of :

Accepted Answer

$Δ = | \begin{matrix} 1 & - 1 & 1 \\ 2 & 2 & α \\ 3 & - 1 & 4 \end{matrix} | = 0$

$\Rightarrow \alpha = 4$

${\Delta _3} = 0$

$= | \begin{matrix} 1 & - 1 & 5 \\ 2 & 2 & 8 \\ 3 & - 1 & β \end{matrix} | = 0$

$\Rightarrow \beta = 14$

$\therefore$ ${x^2} - 18x + 56 = 0$

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