If the system of equations $$ \begin{aligned} & x+y+z=5 \\ & x+2 y+3 z=9 \\ & x+3 y+\alpha z=\beta \end{aligned} $$ has infinitely many solutions, then $\beta-\alpha$ equals:

JEE Main · 2019 · 10 Jan 2019 · Shift 1 · Matrices

If the system of equations

x + y + z = 5 x + 2 y + 3 z = 9 x + 3 y + α z = β

has infinitely many solutions, then

β - α

equals:

Answer: (B)

Infinitely many solutions require the coefficient determinant to vanish. Expanding

111123 13 α = 0

gives

α = 5

. Consistency then needs the augmented determinant to vanish too, and working that condition through gives

β = 13

. Hence

β - α = 13 - 5 = 8

JEE Main · 2019 · 10 Jan 2019 · Shift 1 · Single correct option