If $(3+4 i)^{2025}=5^{2023}(x+i y)$, then $\sqrt{x^2+y^2}=$

If $\left(\frac{\cos \theta+i \sin \theta}{\sin \theta+i \cos \theta}\right)^{2024}+\left(\frac{1+\cos \theta+i \sin \theta}{1-\cos \theta+i \sin \theta}\right)^{2025}=x+i y$ then the value of $x+y$ at $\theta=\frac{\pi}{2}$ is

If $a \pm i b$ and $b \pm a i$ are the roots of $x^4-10 x^3+50 x^2-130 x+169=0$, then $\frac{a}{b}+\frac{b}{a}=$

If $i=\sqrt{-1}$, then $\sum\limits_{n=2}^{30} i^n+\sum\limits_{n=30}^{65} i^{n+3}=$