The roots of the equation $x^3-3x-2=0$ are

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

If $x = \sqrt{6 + \sqrt{6 + \sqrt{6 + \dots \infty}}}$,then which of the following is true?

Which of the following is not a root of $f(x) = x^3 - 11x^2 + 36x - 36$?

For how many values of $k$ is the equation $(1 + 2k)x^2 + (1 - 2k)x + (1 - 2k) = 0$ a perfect square?

Let $R^2$ denote $R \times R$. Let $S = \{(a, b, c) : a, b, c \in R \text{ and } ax^2 + 2bxy + cy^2 > 0 \text{ for all } (x, y) \in R^2 - \{(0, 0)\}\}$. Then which of the following statements is (are) $TRUE$?
$(A) (2, \frac{7}{2}, 6) \in S$
$(B) \text{If } (3, b, \frac{1}{12}) \in S, \text{ then } |2b| < 1$
$(C) \text{For any given } (a, b, c) \in S, \text{ the system of linear equations } ax + by = 1, bx + cy = -1 \text{ has a unique solution.}$
$(D) \text{For any given } (a, b, c) \in S, \text{ the system of linear equations } (a+1)x + by = 0, bx + (c+1)y = 0 \text{ has a unique solution.}$