If the system of linear equations \(3 x+y+\beta z=3\) \(2 \mathrm{x}+\alpha \mathrm{y}-\mathrm{z}=-3\) \(\mathrm{x}+2 \mathrm{y}+\mathrm{z}=4\) has infinitely many solutions, then the value of \(22 \beta-9 \alpha\) is :

JEE Main · 2025 · 02 Apr 2025 · Shift 1 · Matrices

If the system of linear equations

3 x + y + β z = 3

2 x + α y - z = - 3

x + 2 y + z = 4

has infinitely many solutions, then the value of

22 β - 9 α

is :

Answer: (B)

For a system to have infinitely many solutions, both the coefficient determinant

Δ

and all augmented determinants must be zero.

The coefficient determinant:

Δ = 321 1 α 2 β - 1 1 = 0

Expanding:

3 α + 4 β - α β + 3 = 0

... (1)

For the third column augmented determinant:

Δ_{3} = 321 1 α 2 3 - 3 4 = 0

Expanding:

9 α + 19 = 0

, so

α = - \frac{19}{9}

.

Substituting into equation (1):

3 (- \frac{19}{9}) + 4 β - (- \frac{19}{9}) β + 3 = 0

Solving gives

β = \frac{6}{11}

.

Therefore:

22 β - 9 α = 22 \cdot \frac{6}{11} - 9 \cdot (- \frac{19}{9}) = 12 + 19 = 31

JEE Main · 2025 · 02 Apr 2025 · Shift 1 · Single correct option