If `ax^(2)+bx+c=0`, `a ne 0`, `a`, `b`, `c in R` has distinct real roots in `(1,2)`, then `a` and `5a+2b+c` have (a) same sign (b) opposite sign (c) not determined (d) none of these

Options

A. same sign
B. opposite sign
C. not determined
D. none of these

Answer

A

Explanation

`(a)` Let `x_(1)` and `x_(2)` be two roots of `ax^(2)+bx+c=0`
`1 lt x_(1) lt 2` and `1 lt x_(2) lt 2`
Now, `a(5a+2b+c)=a^(2)(5+2(b)/(a)+(c )/(a))`
`=a^(2)(5+2(-1)(x_(1)+x_(2))+x_(1)x_(2))`
`=a^(2){(x_(1)-2)(x_(2)-2)+1} gt 0`
Hence, `a` and `5a+2b+c` are of same sign.