An ordered pair (α , β) for which the system of linear equations (1 + α )x + β y + z = 2 α x+(1+ β )y+z = 3 α x+ β y+2z = 2 has a unique solution is

Question

An ordered pair (α , β) for which the system of linear equations (1 + α )x + β y + z = 2 α x+(1+ β )y+z = 3 α x+ β y+2z = 2 has a unique solution is (A) (1, - 3) (B) (-3,1) (C) (2, 4) (D) (-4, 2)

Tardigrade Admin · Accepted Answer

For unique solution Δ ≠0 ⇒ |1+α&β&1 α&1+β&1 α&β&2| ≠0 |1&-1&0 0&1&-1 α&β&2| ≠0 ⇒ α+β ≠- 2

Q. An ordered pair $(α, β)$ for which the system of linear equations
$(1 + α) x + β y + z = 2$
$αx + (1 + β) y + z = 3$
$αx + β y + 2 z = 2$ has a unique solution is

(1, - 3)

(-3,1)

(2, 4)

(-4, 2)

For unique solution
$Δ \neq = 0 \Rightarrow ∣ ∣ 1 + α α α β 1 + β β 112 ∣ ∣ \neq = 0$
$∣ ∣ 10 α - 1 1 β 0 - 1 2 ∣ ∣ \neq = 0 \Rightarrow α + β \neq = - 2$

Q. An ordered pair (α,β) for which the system of linear equations (1+α)x+βy+z=2 αx+(1+β)y+z=3 αx+βy+2z=2 has a unique solution is