Which of the following values of `alpha` satisfying the equation `|(1+alpha)^2(1+2alpha)^2(1+3alpha)^2(2+alpha)^2(2+2alpha)^2(2+3alpha)^2(3+alpha)^2(3+2alpha)^2(3+3alpha)^2|=-648alpha?` `-4` b. `9` c. `-9` d. `4`

Options

A. `-4`
B. `9`
C. `-9`
D. 4

Answer

2,3

Explanation

Wen have `|{:((1+alpha^(2)),,(1+2alpha)^(2),,(1+3alpha)^(2)),((2+alpha)^(2),,(2+2alpha)^(2),,(2+3alpha)^(2)),((3+alpha)^(2),,(3+2alpha)^(2),,(3+3alpha)^(2)):}|=-648alpha`
Applying `R_(3)to R_(3) -R_(2),R_(2) to R_(2)-R_(1)`
`rArr |{:((1+alpha)^(2),,(1+2alpha)^(2),,(1+3alpha)^(2)),(3+2alpha,,3+4alpha,,3+6alpha),(5+2alpha,,5+4alpha,,5+6alpha):}|=-648alpha`
Applying `R_(3) to R_(3)-R_(2)`
`rArr |{:((1+alpha)^(2),,(1+2alpha)^(2),,(1+3alpha)^(2)),(3+2alpha,,3+4alpha,,3+6alpha),(2,,2,,2):}|=-648alpha`
Applying `C_(3)to C_(3)-C_(2),C_(2)to C_(2)-C_(1)`
`|{:((1+alpha)^(2),,alpha(2+3alpha),,alpha(2+5alpha)),(3+2alpha,,2alpha,,2alpha),(2,,0,,0):}|=-648alpha`
`rArr 2alpha^(2)(2+3alpha)-2alpha^(2)(2+5alpha)=-324alpha`
`rArr -4alpha^(3)=-324alpha`
`rArr alpha=0,+-9`