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. of same type
B. of opposite type
C. undetermined
D. none of these

Answer

A

Explanation

Let `alpha , beta` be the roots of `ax^(2)+bx+c = 0`. Then, `alpha + beta = -(b)/(a) and alpha beta = (c)/(a)`
Now, `a(5a+2b+c) = a^(2)(5+2(b)/(a)+(c)/(a))`
`rArr" "a(5a+2b+c)=a^(2){5-2(alpha+beta)+alpha beta}`
`rArr" "a(5a+2b+c)=a^(2){alpha beta - 2(alpha + beta)+4 + 1}`
`rArr" "a(5a+2b+c)=a^(2){(alpha-2)(beta-2)+1} gt 0`
`rArr" "a(5a + 2b + c) gt 0" "[because alpha, beta lt 2 therefore (alpha-2)(beta-2) gt 0]`
Signs of and 5a + 2b + c are same.