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Question

If |z1|=1, |z2|=2, |z3|=3 and |9z1z2+4z1z3+z2z3|=12 then the value of |z1+z2+z3| is equal to

A
2
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B
3
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C
4
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D
6
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Solution

The correct option is A 2
z¯z=|z|2
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯z1+z2+z3=¯z1+¯z2+¯z3
|z1+z2+z3|2=(z1+z2+z3)(¯z1+¯z2+¯z3)
As |z3|=3 we have z3¯z3=9
z2¯z2=4 and z1¯z1=1
Now, we can write |9z1z2+4z1z3+z2z3|=12
as |z3¯z3z1z2+z2¯z2z1z3+z1¯z1z2z3|=12
|z1z2z3||¯z1+¯z2+¯z3|=12
|¯z1+¯z2+¯z3|=12|z1z2z3|=121×2×3=2


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