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An ordered pair (alpha, beta) for which ...

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

A

(2, 4)

B

`(-4, 2)`

C

`(1, -3)`

D

`(-3, 1)`

Text Solution

Verified by Experts

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).
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