For α, β ∈ 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

Question

For α, β ∈ 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 (A) x2-18 x+56=0 (B) x2+14 x+24=0 (C) x2-10 x+16=0 (D) x2+18 x+56=0

Tardigrade Admin · Accepted Answer

|1 -1 1 2 2 α 3 -1 4|=0 ; 8+α-2(-4+1)+3(-α-2)=0 8+α+6-3 α-6=0 α=4

Q. 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

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

$x^{2} + 14 x + 24 = 0$

$x^{2} - 10 x + 16 = 0$

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

$∣ ∣ 123 - 1 2 - 1 1 α 4 ∣ ∣ = 0; 8 + α - 2 (- 4 + 1) + 3 (- α - 2) = 0$
$8 + α + 6 - 3 α - 6 = 0$
$α = 4$

Q. For α,β∈R, suppose the system of linear equations x−y+z=5 2x+2y+αz=8 3x−y+4z=β has infinitely many solutions. Then α and β are the roots of

x2−18x+56=0

x2+14x+24=0

x2−10x+16=0

x2+18x+56=0