$I. 3x^2 - 20x - 32 = 0$
$II. 2y^2 - 3y - 20 = 0$

If the roots of $x^2 - 7x + 6 = 0$ are $\alpha$ and $\beta$,then $\frac{1}{\alpha} + \frac{1}{\beta} = $

The roots of the equation $ix^2 - 4x - 4i = 0$ are

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 equation $\sqrt{x + 1} - \sqrt{x - 1} = \sqrt{4x - 1}$ has

Let $\alpha$ and $\beta$ be the roots of the equation $px^2 + qx + r = 0$ (where $p \neq 0$). If $p, q, r$ are in $A.P.$ and $\frac{1}{\alpha} + \frac{1}{\beta} = 4$,then the value of $|\alpha - \beta|$ is