The area of a parallelogram whose diagonals are the vectors $2 \bar{a}-\bar{b}$ and $4 \bar{a}-5 \bar{b}$, where $\bar{a}$ and $\bar{b}$ are unit vectors forming an angle of $45^{\circ}$ is

$$ \lim\limits_{x \rightarrow \infty}\left(\frac{x+8}{x+1}\right)^{x+5}=\ldots $$

$$ \cot ^{-1}\left(2 \cos \left(2 \operatorname{cosec}^{-1}(\sqrt{2})\right)\right)=\ldots $$

If $\mathrm{z}=x+\mathrm{i} y$ is a complex number, then the equation $\left|\frac{z+i}{z-i}\right|=\sqrt{3}$ represents the