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Which of the following is correct for an...

Which of the following is correct for any tow complex numbers `z_1a n dz_2?` `|z_1z_2|=|z_1||z_2|` (b) `a r g(z_1z_2)=a r g(z_1)a r g(z_2)` (c) `|z_1+z_2|=|z_1|+|z_2|` (d) `|z_1+z_2|geq|z_1|+|z_2|`

A

`|z_(1) z_(2)| = | z_(1) ||z_(2)|`

B

arg` (z _(1) z_(2)) = arg (z_(1). Arg (z_(2))`

C

`|z_(1) + z_(2)| = |z_(1)|+|z_(2)|`

D

`|z_(1) + z_(2) ge |z_(1)|- | z_(2)|`

Text Solution

Verified by Experts

The correct Answer is:
A
Let `z_(1) = r_(1) (costheta_(1) + isintheta_(1))`
`rArr |z_(1)| = r-(1) …(i) `
and `z_(2) = r_(2)(costheta_(2) + isin theta_(2)) `
`rArr |z_(2)| =r_(2) ...(ii)`
Now. ` z_(1) z_(2) = r_(1) r_(2) [costheta_(1) costheta_(2) + isin theta_(1) costheta_(2)+ icostheta_(1)isintheta_(2)+i^(2) sintheta_(1)sintheta_(2)]`
`=r_(1)r_(2) [cos(theta_(1) + theta_(2))+ isin(theta_(1) + theta_(2))]`
`rArr |z_(1)z_(2)| = r_(1)r_(2)`
`:. |z_(1)z_(2)| = |z_(1)||z_(1)| [using Eqa. (i) and (ii)]`
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