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Let $z$ be a complex number such that $|z|+z=3+i$,where $i=\sqrt{-1}$. Then $|z|$ is equal to:

If $\alpha$ and $\beta$ are different complex numbers with $|\beta|=1,$ then find $\left|\frac{\beta-\alpha}{1-\bar{\alpha} \beta}\right|.$

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The complex number $\frac{1+2i}{1-i}$ lies in

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

If $a > 0$ and $z = \frac{(1 + i)^2}{a - i}$,where $i = \sqrt{-1}$,has magnitude $\sqrt{\frac{2}{5}}$,then $\overline{z}$ is equal to

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