For $\alpha, \beta \in \mathbb{R}$, suppose the system of linear equations $\mathbf{x}-y+z=5$ $2 \mathrm{x}+2 \mathrm{y}+\alpha \mathrm{z}=8$ $3 x-y+4 z=\beta$ has infinitely many solutions. Then $\alpha$ and $\beta$ are the roots of

(A) $\mathrm{x}^{2}-10 \mathrm{x}+16=0$

JEE Main · 2023 · 30 Jan 2023 · Shift 2 · Matrices

For

α, β \in R

, suppose the system of linear equations

x - y + z = 5

2 x + 2 y + α z = 8

3 x - y + 4 z = β

has infinitely many solutions. Then

α

and

β

are the roots of

Answer: (C)

Infinitely many solutions require the coefficient determinant to vanish:

123 - 1 2 - 1 1 α 4 = 1 (8 + α) + (8 - 3 α) - 8 = 8 - 2 α = 0

, so

α = 4

. Expressing the first two equations in terms of a free parameter and substituting into

3 x - y + 4 z

yields the constant

14

, so consistency demands

β = 14

. With

α + β = 18

and

α β = 56

, these are the roots of

x^{2} - 18 x + 56 = 0

JEE Main · 2023 · 30 Jan 2023 · Shift 2 · Single correct option