Discussion
If \(z\,|z| = 24 + 7i\), where z is a complex number, then the value of |z| is equal to
Answer: A
|z| = 5.
Take the modulus of both sides. Since |z| is a real number, \(|z\,|z|| = |z| \cdot |z| = |z|^{2}\).
\[ \begin{aligned} |z|^{2} &= |24 + 7i| = \sqrt{576 + 49} \\ &= \sqrt{625} = 25 \\ |z| &= 5 \end{aligned} \]
25 is \(|z|^{2}\), the trap of stopping one step early. Check: z = \(\frac{24 + 7i}{5}\) has modulus 5, and 5z = 24 + 7i.
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