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

If $x$ is a solution of the equation $\sqrt{2x + 1} - \sqrt{2x - 1} = 1$ where $x \ge \frac{1}{2}$,then $\sqrt{4x^2 - 1}$ is equal to:

Let $S$ be the set of all real roots of the equation,$3^{x}(3^{x}-1)+2=|3^{x}-1|+|3^{x}-2|$. Then $S$

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

Solve the given two equations and select the correct answer from the given options.
$I.$ $7x - 3y = 13$
$II.$ $5x + 4y = 40$

If $\alpha$ is a root of the quadratic equation $x^2 + 6x - 2 = 0$,then another root $\beta$ is