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

Options

A. (2, 4)
B. `(-4, 2)`
C. `(1, -3)`
D. `(-3, 1)`

Explanation

Given system of linear equations,
`(1+alpha)x +betay +z = 2`
`alphax +(1+beta)y +z =3`
`alphax +betay +2z =2`
has a unique solution, if
`|{:(1+alpha, beta, 1), (alpha, (1+beta), 1), (alpha, beta, 2):}| ne 0`
Applying `R_(1) to R_(1) - R_(3) " and " R_(2) to R_(2)-R_(3)`
`|{:(1, 0, -1), (0, 1, -1), (alpha, beta, 2):}| ne 0`
`rArr 1(2+beta)-0(0 + alpha)-1(0-alpha) ne 0`
`rArr alpha +beta +2 ne 0 " "...(i)`
Note that, only (2, 4) satisfy the Eq. (i).