For the quadratic equation `x^2+2(a+1)x+9a-5=0,` which of the following is/are true? (a) If `2 < a < 5` , then roots are opposite sign (b)If `a < 0` , then roots are opposite in sign (c) if `a > 7` then both roots are negative (d) if `2 <= a <= 5` then roots are unreal

Options

A. If ` 2 lt a lt 5` , then roots are of opposite sign .
B. If ` a lt 0, ` then roots are of opposite sign
C. If ` a gt 7` , then both roots are negative .
D. If `2 le a le 5` , then roots are unreal.

Answer

2,3,4

Explanation

Given equation is
`x^(2) + 2 (a + 1) x + 9 a - 5 = 0 `
` D = 4 (a + 1)^(2) - 4 (9a - 5) = 4 (a -1) (a - b)`
` D ge 0 rArr a le 1 or a ge 6`
If ` a lt 0 `, then ` 9a - 5 lt 0 ` . Hence, the products of roots is less
then 0.So, the roots are of opposite sign .If ` a gt 7`, then sum of
roots is ` - 2(a + 1) lt 0 ` . Product of roots is greater then 0