The quadratic polynomial $p(x)$ has roots $1$ and $\alpha$, while quadratic polynomial $q(x)$ has roots $1$ and $\beta$. Let $\alpha$ and $\beta$ be the roots of $r(x) = p(x) + q(x)$. Then $\lim_{x\to\infty}\left[\sqrt{p(x)} - \sqrt{q(x)}\right] =$

If $x = \sqrt{-1-\sqrt{-1-\sqrt{-1-\ldots\infty}}}$, where $\omega$ is a non-real complex cube root of unity, then the value of $x$ is...

If $\sin 3\alpha = 4\sin\alpha \cdot \sin(x+\alpha) \cdot \sin(x-\alpha)$ where $\alpha \neq n\pi, n \in Z$, then all possible values of $x$ are given as

If $\cos(\theta-\alpha)=a$ and $\sin(\theta-\beta)=b$, then the value of $\cos^2(\alpha-\beta)+2ab\sin(\alpha-\beta)+\cos^2(\theta-\alpha)$ is...