Question:
Which of the following is correct for any two complex numbers z1 and z2?
(a) $\left|z_{1} z_{2}\right|=\left|z_{1}\right|\left|z_{2}\right|$
(b) $\arg \left(z_{1} z_{2}\right)=\arg \left(z_{1}\right) \arg \left(z_{2}\right)$
(c) $\left|z_{1}+z_{2}\right|=\left|z_{1}\right|+\left|z_{2}\right|$
(d) $\left|z_{1}+z_{2}\right| \geq\left|z_{1}\right|+\left|z_{2}\right|$
Solution:
Since, it is known that
$\left|z_{1} z_{2}\right|=\left|z_{1}\right|\left|z_{2}\right|$
$\arg \left(z_{1} z_{2}\right)=\arg \left(z_{1}\right)+\arg \left(z_{2}\right)$ and
$\left|z_{1}+z_{2}\right| \leq\left|z_{1}\right|+\left|z_{2}\right|$
Hence, the correct option is (a) .