The number of integral values of m for which the quadratic expression 1 + 2 m x 2 - 2 1 + 3 m x + 4 1 + m , x ∈ R is always positive, is

Step-by-step solution

∵ Quadratic expression is positive, hence 1 + 2 m > 0 and D < 0 ⇒ m > - 1 2 and 4 1 + 3 m 2 - 16   1 + m 1 + 2 m < 0 ⇒ 9 m 2 + 1 + 6 m - 4 2 m 2 + 3 m + 1 < 0 ⇒ m 2 - 6 m - 3 < 0 ⇒ m - 3 + 2 3 m - 3 - 2 3 < 0 ⇒ m ∈ 3 - 2 3 ,   3 + 2 3 ∴ m = 0 ,   1 ,   2 ,   3 ,   4 ,   5 ,   6 ∴ Number of integral values = 7