Question

For any two non-zero complex numbers $Z_1$ and $Z_2$, the value of $(\left|Z_1\right|+\left|Z_2\right|)|\frac{Z_1}{\left|Z_1\right|}+\frac{Z_2}{\left|Z_2\right|}|$ is:

• A. less than $2(|Z_1|+|Z_2|)$
• B. greater than $2(|Z_1|+|Z_2|)$
• C. greater than or equal to $2(|Z_1|+|Z_2|)$
• D. less than or equal to $2(|Z_1|+|Z_2|)$

Answer 1

The correct option is D less than or equal to $2(|Z_1|+|Z_2|)$

$|\frac{Z_1}{|Z_1|}+\frac{Z_2}{|Z_2|}|\le \left|\frac{Z_1}{|Z_1|}\right|+\left|\frac{Z_2}{|Z_2|}\right|\le 2$
$\therefore |\frac{Z_1}{|Z_1|}+\frac{Z_2}{|Z_2|}|\le 2$
Multiplying both sides by $(|Z_1|+|Z_2|)$, we get
$(|Z_1|+|Z_2|)|\frac{Z_1}{|Z_1|}+\frac{Z_2}{|Z_2|}|\le 2(|Z_1|+|Z_2|)$