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Answer 1

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Step-by-step explanation:

$so \: here \: \\ for \: a \: certain \: \: complex \: number \\ z = 2 - i \\ w = 1 + 3i \\ \\ hence \: then \\ \frac{z {}^{2} }{w - i} = a + ib \\ \\ = \frac{(2 - i) {}^{2} }{1 + 3i - i} \\ \\ = \frac{4 + i {}^{2} - 4i}{ 1 + 2i} \\ \\ = \frac{4 + ( - 1) - 4i}{1 + 2i} \\ \\ = \frac{3 - 4i}{1 + 2i} \\ \\ = \frac{3 - 4i}{1 + 2i} \times \frac{1 - 2i}{1 - 2i} \\ \\ = \frac{(3 - 4i)(1 - 2i)}{(1) {}^{2} - (2i) {}^{2} } \\ \\ = \frac{3 - 6i - 4i + 8i {}^{2} }{1 - 4i {}^{2} } \\ \\ = \frac{3 - 10i + 8( - 1)}{1 - 4( - 1)} \\ \\ = \frac{3 - 8 - 10i}{1 + 4} \\ \\ = \frac{ - 5 - 10i}{5} \\ \\ = \frac{5( - 1 - 2i)}{5} \\ \\ = - 1 - 2i \\ \\ comparing \: real \: and \: imaginary \: parts \\ a = - 1 \\ b = - 2$

$we \: know \: that \\ |z| = \sqrt{a {}^{2} + b {}^{2} } \\ \\ = \sqrt{( - 1) {}^{2} + ( - 2) {}^{2} } \\ \\ = \sqrt{1 + 4} \\ \\ |z| = \sqrt{5}$