Let $S = \{ x : x \in R \text{ and } (\sqrt{3} + \sqrt{2})^{x^2 - 4} + (\sqrt{3} - \sqrt{2})^{x^2 - 4} = 10 \}$. Then $n(S)$ is equal to

For what value of $a$ does the equation $(a^2 - a - 2)x^2 + (a^2 - 4)x + a^2 - 3a + 2 = 0$ have more than two solutions?

If the roots of the equation $(m - n)x^2 + (n - l)x + (l - m) = 0$ are equal,then which of the following is true for $l, m,$ and $n$?

Let $p, q$ and $r$ be real numbers $(p \ne q, r \ne 0)$ such that the roots of the equation $\frac{1}{x + p} + \frac{1}{x + q} = \frac{1}{r}$ are equal in magnitude but opposite in sign. Then the sum of squares of these roots is equal to:

Let $\alpha$ and $\beta$ be the roots of the quadratic equation $x^2 \sin \theta - x(\sin \theta \cos \theta + 1) + \cos \theta = 0$ where $0 < \theta < 45^\circ$,and $\alpha < \beta$. Then $\sum_{n=0}^\infty (\alpha^n + \frac{(-1)^n}{\beta^n})$ is equal to

The solution of the equation $\sqrt{x + 10} + \sqrt{x - 2} = 6$ is