Let $P Q R S$ be a quadrilateral in a plane, where

$Q R=1, \angle P Q R=\angle Q R S=70^{\circ}, \angle P Q S=15^{\circ}$ and $\angle P R S=40^{\circ}$.

If $\angle R P S=\theta^{\circ}, P Q=\alpha$ and $P S=\beta$, then the interval(s) that contain(s) the value of

$4 \alpha \beta \sin \theta^{\circ}$ is/are

Let

$$ \alpha=\sum\limits_{k = 1}^\infty {{{\sin }^{2k}}\left( {{\pi \over 6}} \right)} $$

Let $g:[0,1] \rightarrow \mathbb{R}$ be the function defined by

$$ g(x)=2^{\alpha x}+2^{\alpha(1-x)} . $$

Then, which of the following statements is/are TRUE ?

Let $\bar{z}$ denote the complex conjugate of a complex number $z$. If $z$ is a non-zero complex number for which both real and imaginary parts of $$ (\bar{z})^{2}+\frac{1}{z^{2}} $$ are integers, then which of the following is/are possible value(s) of $|z|$ ?

Let $G$ be a circle of radius $R>0$. Let $G_{1}, G_{2}, \ldots, G_{n}$ be $n$ circles of equal radius $r>0$. Suppose each of the $n$ circles $G_{1}, G_{2}, \ldots, G_{n}$ touches the circle $G$ externally. Also, for $i=1,2, \ldots, n-1$, the circle $G_{i}$ touches $G_{i+1}$ externally, and $G_{n}$ touches $G_{1}$ externally. Then, which of the following statements is/are TRUE?