Question

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

Answer 1

Given determinant could be expressed as product of two determinants.
i.e., `|{:((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)):}|= -648 alpha`
` rArr |{:(1+2alpha+alpha^(2), 1+4alpha+4alpha^(2), 1+6alpha+9alpha^(2)), (4+4alpha+alpha^(2), 4+8alpha+4alpha^(2), 4+12alpha+9alpha^(2)), (9+6alpha+alpha^(2), 9+12alpha+4alpha^(2), 4+18alpha+9alpha^(2)):}|= -648 alpha`
`rArr|{:(1, alpha, alpha^(2)), (4, 2alpha, alpha^(2)), (9, 3alpha, alpha^(2)):}| * |{:(1, 1, 1), (2, 4, 6), (1, 4, 9):}|= -648 alpha`
`rArr alpha^(3)|{:(1, 1, 1), (4, 2, 1), (9, 3, 1):}|*|{:(1, 1, 1), (2, 4, 6), (1, 4, 9):}|= -648 alpha`
`rArr -8 alpha^(3) = -648 alpha`
`rArr alpha^(3) - 81alpha = 0 rArr alpha (alpha^(2)-81) = 0`
`therefore alpha = 0, +-9`