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Question

If z1,z2,z3 are imaginary numbers such that |z1|=|z2|=|z3|=1z1+1z2+1z3=1 then |z1+z2+z3| is

A
Equal to 1
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B
Less than 1
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C
Greater than 1
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D
Equal to 3
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Solution

The correct option is A Equal to 1
using z¯¯¯z=|z|2
z1¯¯¯¯¯z1=|z1|2=11z1=¯¯¯¯¯z1
z2¯¯¯¯¯z2=|z2|2=11z2=¯¯¯¯¯z2
and z¯¯¯¯¯z3=|z3|2=11z3=¯¯¯¯¯z3

Now using, 1z1+1z2+1z3=1
|¯¯¯z1+¯¯¯z2+¯¯¯z3|=1
|¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯z1+z2+z3|=1
|z1+z2+z3|=1, since |¯¯¯z|=|z|

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