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Question
Which of the following is correct for any two complex numbers z1 and z2?
Options
|z1z2| = |z1||z2|
arg(z1z2) = arg(z1).arg(z2)
|z1 + z2| = |z1| + |z2|
|z1 + z2| ≥ |z1| – |z2|
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Solution
|z1z2| = |z1||z2|
Explanation:
Let z1 = r1(cosθ1 + isinθ1)
∴ |z2| = r1
And z2 = r2(cosθ2 + isinθ2)
∴ |z2| = r2
z1z2 = r1(cosθ1 + isinθ1).r2(cosθ2 + isinθ2)
= r1r2(cosθ1 + isinθ1).(cosθ2 + isinθ2)
= r1r2(cosθ1 cosθ2 + isinθ2 cosθ1 + isinθ1 cosθ2 + i2sinθ1 sinθ2)
= r1r2 [(cosθ1 cosθ2 – sinθ1 sinθ2) + i(sinθ1 cosθ2 + cosθ1 sinθ2)]
= r1r2 [cos(θ1 + θ2) + isin(θ1 + θ2)]
∴ |z1z2| = |z1||z2|
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