Let `a`, `b`, `c` and `m in R^(+)`. The possible value of `m` (independent of `a`, `b` and `c`) for which atleast one of the following equations have real roots is
`{:(ax^(2)+bx+cm=0),(bx^(2)+cx+am=0),(cx^(2)+ax+bm=0):}}`

Options

A. `(1)/(2)`
B. `(1)/(8)`
C. `(1)/(12)`
D. `(1)/(4)`

Answer

B, C, D

Explanation

`(b,c,d)` If at least one of the equations has real roots, then
`D_(1)+D_(2)+D_(3) ge 0`
`(b^(2)-5acm)+(c^(2)-4bam)+a^(2)-4cbm ge 0`
`a^(2)+b^(2)+c^(2) ge 4(ab+bc+ca)m`
`4m ge (a^(2)+b^(2)+c^(2))/(ab+bc+ca)`………`(1)` `AA a,b,c in R+`
but `a^(2)+b^(2) ge 2ab` etc.
`:. a^(2)+b^(2)+c^(2) ge ab+bc+ca`
`(a^(2)+b^(2)+c^(2))/(ab+bc+ca) ge 1`
`:.(a^(2)+b^(2)+c^(2))/(ab+bc+ca)|_(min)=1` , Hence `4m` must be less than or equal to the minimum value.
`:. 4m le 1` gtbrgt `implies m le (1)/(4)`
`implies m in (0,(1)/(4)]`