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Question

An ordered pair (α,β) for which the system of linear equations
(1+α)x+βy+z=2αx+(1+β)y+z=3αx+βy+2z=2
has a unique solution, is:
  1. (1,3)
  2. (3,1)
  3. (4,2)
  4. (2,4)


Solution

The correct option is D (2,4)
The system of linear equation are
(1+α)x+βy+z=2αx+(1+β)y+z=3αx+βy+2z=2
for unique solution, Δ0
∣ ∣1+αβ1α1+β1αβ2∣ ∣0
Apply R1R1R2
∣ ∣110α1+β1αβ2∣ ∣0
1(2+2ββ)+1(2αα)+0 0α+β+2 0α+β2
Hence, (2,4) will satisfy this condition.

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