In a triangle $PQR$,$\angle R = \frac{\pi}{2}$. If $\tan(\frac{P}{2})$ and $\tan(\frac{Q}{2})$ are the roots of the equation $ax^2 + bx + c = 0$ $(a \neq 0)$,then:

Let $\alpha, \beta$ with $\alpha > \beta$ be the roots of the equation $x^2 - \sqrt{2}x - \sqrt{3} = 0$. Let $P_n = \alpha^n - \beta^n$ for $n \in \mathbb{N}$. Then $(11\sqrt{3} - 10\sqrt{2})P_{10} + (11\sqrt{2} + 10)P_{11} - 11P_{12}$ is equal to:

If one real root of the quadratic equation $81x^2 + kx + 256 = 0$ is the cube of the other root,then a value of $k$ is

Let $\alpha$ and $\beta$ be the roots of the equation $px^2+qx-r=0$,where $p \neq 0$. If $p, q,$ and $r$ are the consecutive terms of a non-constant $G$.$P$. and $\frac{1}{\alpha}+\frac{1}{\beta}=\frac{3}{4}$,then the value of $(\alpha-\beta)^2$ is:

If $\alpha^2 = 5\alpha - 3$ and $\beta^2 = 5\beta - 3$ where $\alpha \neq \beta$,what is the value of $\frac{\alpha}{\beta} + \frac{\beta}{\alpha}$?

If $\sin 2\theta$ and $\cos 2\theta$ are solutions of $x^2 + ax - c = 0$,then