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if the value of the determinant |{:(a,...

if the value of the determinant `|{:(a,,1,,1),(1,,b,,1),(1,,1,,c):}|` is positivie then
`(a,b,c lt 0)`

A

`abc gt 1`

B

` abc gt -8`

C

` abc gt -9`

D

`abc gt -2`

Text Solution

Verified by Experts

The correct Answer is:
B
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 gt 3 (abc)^(1//3)`
`[ :' A.M gt G.M. rArr .(ab+c)/(3) gt (abc)^(1//3)]`
`rArr x^(3)+ 2 gt 3x , " where " x= (abc)^(1//3)`
`rArr x^(3)- 3x+2 gt 0 rArr (x-1)^(2) (x+2) gt 0`
`rArr x+2 gt 0 rArr x gt -2 rArr (abc)^(1//3) gt -2 rArr abc gt-8`
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