If $\alpha, \beta$ are the roots of $x^2 - 3x + a = 0, a \in R$ and $\alpha < 1 < \beta$,then:

Let $a$,$b$,and $c$ be the roots of the equation $x^3 + 8x + 1 = 0$. Then the value of $\frac{bc}{(8b + 1)(8c + 1)} + \frac{ac}{(8a + 1)(8c + 1)} + \frac{ab}{(8a + 1)(8b + 1)}$ is equal to:

Solve the given two equations and select the correct answer from the given options.
$I.$ $x^{2} - 3481 = 0$
$II.$ $3y^{2} = \sqrt[3]{216000}$

The number of solutions of $\log_4(x - 1) = \log_2(x - 3)$ is:

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 $\alpha$ and $\beta$ are the roots of the equation $2x(2x + 1) = 1$,then $\beta$ is equal to