If the lvalue of the determinants `|(a,1,1),(1,b,1),(1,1,c)|` is positive then:

Options

A. `abc gt 1`
B. `abc gt -8`
C. `abc lt -8`
D. `abc gt -2`

Answer

B

Explanation

We have
`Delta = |(a,1,1),(1,b,1),(1,1,c)| = abc - (a +b +c) +2`
`:. Delta gt 0`
`rArr abc + 2 gt a + b + c`
`rArr abc + 2 a + b + c`
`rArr abc + 2 gt 3 (abc)^(1//3) [ :' A.M. gt G.M. rArr (a +b +c)/(3) gt (abc)^(1//3)]`
`x^(3) + 2 gt 3x`, where `x = (abc)^(1//3)`
`rArr x^(3) - 3x + 2 gt 0`
`rArr (x -1)^(2) (x +1) gt 0`
`rArr x +2 gt 0 rArr x gt -2 rArr (abc)^(1//3) gt -2 rArr abc gt -8`