Q.
Consider, $f ( z )= z ^{12}+2 \cdot z ^{11}+3 \cdot z ^{10}+\ldots . .+12 \cdot z +13$ and $\alpha=\cos \frac{2 \pi}{13}+ i \sin \frac{2 \pi}{13}$, where $i =\sqrt{-1}$.
If $S=\left\{ z \mid \operatorname{Re}\left( f ( z )- z \left(\frac{ z ^{13}-1}{( z -1)^2}\right)\right)=\frac{13}{4}\right\}$, then maximum area of the quadrilateral formed by joining four points lying in $S$ is
[Note: $\operatorname{Re}(z)$ denote the real part of complex number of $z$.
Complex Numbers and Quadratic Equations
Solution: