The ordered pair (a, b), for which the system of linear equations $3 \mathrm{x}-2 \mathrm{y}+\mathrm{z}=\mathrm{b}$ $5 \mathrm{x}-8 \mathrm{y}+9 \mathrm{z}=3$ $2 \mathrm{x}+\mathrm{y}+\mathrm{az}=-1$ has no solution, is :

(C) $\left(-3,-\frac{1}{3}\right)$

(A) $\left(3, \frac{1}{3}\right)$

(B) $\left(-3, \frac{1}{3}\right)$

(D) $\left(3,-\frac{1}{3}\right)$

JEE Main · 2022 · 26 Jun 2022 · Shift 1 · Matrices

The ordered pair (a, b), for which the system of linear equations

3 x - 2 y + z = b

5 x - 8 y + 9 z = 3

2 x + y + az = - 1

has no solution, is :

Answer: (C)

For no solution, the coefficient determinant

Δ = 0

but at least one of the determinants formed by replacing a column with the constant terms is non-zero.

Compute

Δ = 352 - 2 - 8 1 19 a = 3 (- 8 a - 9) + 2 (5 a - 18) + 1 (21) = 0

.

Simplifying:

- 24 a - 27 + 10 a - 36 + 21 = 0 \Rightarrow - 14 a - 42 = 0 \Rightarrow a = - 3

.

For no solution, we need

Δ_{2} \neq = 0

(replacing the second column with constants). Testing

b = - \frac{1}{3}

ensures

Δ_{2} \neq = 0

.

Therefore,

(a, b) = (- 3, - \frac{1}{3})

JEE Main · 2022 · 26 Jun 2022 · Shift 1 · Single correct option