Question

Let a, b, c be distinct real number and D be the determinant given by
`D = |(a,1,1),(1,b,1),(1,1,c)|`
Statement1 : If `D gt 0 " then " abc gt - 8`
Statement - 2: `A.M. gt G.M.`.
A. Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 5
B. Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 5
C. Statement 1 is true, Statement 2 is False
D. Statement 1 is False, Statement 2 is true

Answer 1

Correct Answer - A
We have,
`D = |(a,1,1),(1,b,1),(1,1,c)| = abc - (a + b +c) + 2`
`:. D gt 0`
`rArr abc - (a + b +c) + 2 gt 0 `
`rArr abc + 2 gt a + b + c`...(i)
But, A.M. `gt G.M.`
`rArr (a + b +c)/(3) gt (abc) (1//3)`
`rArr a + b + c gt 3 (abc)^(1//3)`...(ii)
From (i) and (ii), we get
`abc + 2 gt 3 (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 (abc)^(1//3) + 2 gt 0`
`rArr (abc)^(1//3) gt -2 rArr abc gt -8`