Question

If `ax^2 + bx+ c = 0`, `a!=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
A. of same type
B. of opposite type
C. undetermined
D. none of these

Answer 1

Correct Answer - A
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.