$I. 6x^2 + 41x + 63 = 0$
$II. 4y^2 + 8y + 3 = 0$

Let $y = f(x) = ax^2 + 2bx + c$ (where $a, b, c \in R$ and $a \neq 0$). If $f(x) = 0$ has imaginary roots and $4a + 4b + c < 0$,then which of the following is true?

If $x = \sqrt{7 + 4\sqrt{3}},$ then $x + \frac{1}{x} = $

The number of real roots of the equation $e^{4x} + e^{3x} - 4e^{2x} + e^x + 1 = 0$ is:

$\sqrt{6+\sqrt{6+\sqrt{6+\cdots}}}$ is equal to

The value of $k$ for which the roots $\alpha, \beta$ of the equation $x^{2}-6x+k=0$ satisfy the relation $3\alpha+2\beta=20$ is: