In a $$\Delta $$PQR = 30$$^\circ $$ and the sides PQ and QR have lengths 10$$\sqrt 3 $$ and 10, respectively. Then, which of the following statement(s) is(are) TRUE?

Let $$\alpha $$ and $$\beta $$ be non zero real numbers such that $$2(\cos \beta - \cos \alpha ) + \cos \alpha \cos \beta = 1$$. Then which of the following is/are true?

Let $$f\left( x \right) = x\sin \,\pi x,\,x > 0.$$ Then for all natural numbers $$n,\,f'\left( x \right)$$ vanishes at

Let $$\theta ,\,\varphi \, \in \,\left[ {0,2\pi } \right]$$ be such that
$$2\cos \theta \left( {1 - \sin \,\varphi } \right) = {\sin ^2}\theta \,\,\left( {\tan {\theta \over 2} + \cot {\theta \over 2}} \right)\cos \varphi - 1,\,\tan \left( {2\pi - \theta } \right) > 0$$ and $$ - 1 < \sin \theta \, < - {{\sqrt 3 } \over 2},$$

then $$\varphi $$ cannot satisfy