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Let alpha and beta be the roots of the e...

Let `alpha` and `beta` be the roots of the equation `x^(2)+x+1=0`. Then for `y ne 0` in R, `|(y+1,alpha,beta),(alpha,y+beta,1),(beta,1,y+alpha)|` is equal to :

A

`y(y^(2)-3)`

B

`y^(3)`

C

`y(y^(2)-1)`

D

`y^(3)-1)`

Text Solution

Verified by Experts

The correct Answer is:
B
Roots of `x^(2)+x+1=0` are `omega, omega^(2)`
`rArr alpha= omega, beta= omega^(2) rArr 1+alpha+beta=0`
`|(y+1, alpha, beta),(alpha,y+beta,1),(beta,1,y+alpha)|=|(y, alpha, beta),(y,y+beta,1),(y,1,y+alpha)|=y|(1,alpha,beta),(0,y+alpha,1-beta),(0,1-alpha,y+alpha-beta)|`
`(C_(1)rarr C_(1)+C_(2)+C_(3))" " (R_(2) rarr R_(2)-R_(1))(R_(3) rarr R_(3)-R_(1))`
`=y|(1,alpha, beta),(0,(y-isqrt(3)),1-beta),(0,1-alpha,(y+i sqrt(3)))|`
`=y{(y^(2)+3)-(1-alpha-beta+alpha beta)}`
`=y{y^(2)+3-(2+1)}=y^(3)`
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