Let z be the complex number such that $|\mathrm{z}|+\mathrm{z}=3+\mathrm{i}$ where $i=\sqrt{-1}$, then $|\mathrm{z}|=$

Let $z = x + iy$ where $x, y \in \mathbb{R}$ with modulus $|z| = \sqrt{x^2 + y^2}$. Given the equation $|z| + z = 3 + i$, substitute to obtain:

$\sqrt{x^2 + y^2} + x + iy = 3 + i$

Equating real and imaginary parts yields $y = 1$ and $\sqrt{x^2 + 1} + x = 3$.

Solving the real equation: $\sqrt{x^2 + 1} = 3 - x$. Squaring both sides gives $x^2 + 1 = 9 - 6x + x^2$, which simplifies to $6x = 8$, so $x = \frac{4}{3}$.

Thus $z = \frac{4}{3} + i$ and $|z| = \sqrt{(\frac{4}{3})^2 + 1^2} = \sqrt{\frac{16}{9} + \frac{9}{9}} = \sqrt{\frac{25}{9}} = \frac{5}{3}$.