Quadratic Equation and Inequalities

If the roots of the equation $b{x^2} + cx + a = 0$ imaginary, then for all real values of $x$, the expression $3{b^2}{x^2} + 6bcx + 2{c^2}$ is :

If $\lambda$ be the ratio of the roots of the quadratic equation in x, 3m²x² + m(m – 4)x + 2 = 0, then the least value of m for which $\lambda + {1 \over \lambda } = 1,$ is

The equation ${e^{\sin x}} - {e^{ - \sin x}} - 4 = 0$ has:

The sum of all the real values of x satisfying the equation 2(x$-$1)(x² + 5x $-$ 50) = 1 is :

The number of all possible positive integral values of $\alpha$  for which the roots of the quadratic equation, 6x² $-$ 11x + $\alpha$ = 0 are rational numbers is :

The number of integral values of m for which the equation (1 + m² )x² – 2(1 + 3m)x + (1 + 8m) = 0 has no real root is :

If m is chosen in the quadratic equation (m² + 1) x² – 3x + (m² + 1)² = 0 such that the sum of its roots is greatest, then the absolute difference of the cubes of its roots is :-

Let f(x) be a quadratic polynomial such that f(–1) + f(2) = 0. If one of the roots of f(x) = 0 is 3, then its other root lies in :

The set of all values of K > $-$1, for which the equation ${(3{x^2} + 4x + 3)^2} - (k + 1)(3{x^2} + 4x + 3)(3{x^2} + 4x + 2) + k{(3{x^2} + 4x + 2)^2} = 0$ has real roots, is :

The numbers of pairs (a, b) of real numbers, such that whenever $\alpha$ is a root of the equation x² + ax + b = 0, $\alpha$² $-$ 2 is also a root of this equation, is :