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  • Which of the following is correct for any two complex numbers z1 and z2? A. |z1 z2| = |z1||z2| B. arg (z1z2) = arg (z1). Arg (z2) C. |z1 + z2| = |z1|+ |z2| D. |z1 + z2| ≥ |z1| – |z2|

Which of the following is correct for any two complex numbers z_1\: \: and\: \: \: \: z_2  ?
A. |z_1 z_2| = |z_1||z_2|
B. arg (z_1z_2) = arg (z_1). Arg (z_2)
C. |z_1 + z_2| = |z_1|+ |z_2|
D. |z_1 + z_2| \geq |z_1| - |z_2|

Answers (1)

The answer is the option (a).

\begin{array}{c} z_{1}=r_{1}\left(\cos \theta_{1}+i \sin \theta_{2}\right) \\ \left|z_{1}\right|=r_{1} \\ z_{2}=r_{2}\left(\cos \theta_{2}+i \sin \theta_{2}\right) \\ \left|z_{2}\right|=r_{2} \\\\ z_1z_2=r_{1} r_{2}\left[\cos \left(\theta_{1}+\theta_{2}\right)+i \sin \left(\theta_{1}+\theta_{2}\right)\right] \\ =r_{1}\left(\cos \theta_{1}+i \sin \theta_{1}\right) \cdot r_{2}\left(\cos \theta_{2}+i \sin \theta_{2}\right) \\ \end{array}

\begin{array}{c} =r_{1} r_{2}\left[\left(\cos \theta_{1} \cos \theta_{2}-\sin \theta_{1} \sin \theta_{2}\right)+i\left(\sin \theta_{1} \cos \theta_{2}+\cos \theta_{1} \sin \theta_{2}\right)\right] \\ =r_{1} r_{2}\left[\cos \left(\theta_{1}+\theta_{2}\right)+i \sin \left(\theta_{1}+\theta_{2}\right)\right]\left|z_{1} z_{2}\right|=\left|z_{1} \| z_{2}\right| \end{array}

Hence,  option  a is correct.

Posted by

infoexpert21

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