Let f (x) = ax2 + bx + c, a, b, c Ïµ R. If f (x) Takes Real Values For Real Values Of x and Non – Real Values For Non – Real Values Of x, Then.

The real values of a for which the quadratic equation 3x² + 2 (a² + 1) x+(a² – 3a + 2) = 0 possesses roots opposite signs lie in

If a, b, c Ïµ R and a + b + c = 0, then the quadratic equation 3 ax² + 3ax² + 2bx + c = 0 has

Let f (x) = ax2 + bx + c, a, c, Ïµ R and a ≠ 0. Suppose f (x) > 0 for all x Ïµ R.

Let g (x) = f (x) + f ’ (x) + f ”(x). Then

If b < 0, then the roots x₁ and x₂ of the equation 2x² + 6x + b = 0, satisfy the condition (x₁/x₂) < k where k is equal to.

If ax² + bx + c, a, b, c Ïµ R has no real zeros, and if c < 0, then,

If x is real, then the maximum value of    

If both the roots of the equation x² – 6ax + 2 – 2a + 9a² = 0 exceed 3, then

Let α, β be the roots of the equation x² – ax + b = 0 and A_n = αⁿ + βⁿ. 

If α, β, γ are such that α + β + γ = 2, α² + β² + γ² = 6, α³ + β³ + γ³ = 8, then α⁴ + β⁴ + γ⁴ is