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If $|z_{1}| = |z_{2}| = |z_{3}| = 1$ and $z_{1} + z_{2} + z_{3} = \sqrt{2} + i$ then the number $z_{1} {\bar{z}}_{2} + z_{2} {\bar{z}}_{3} + z_{3} {\bar{z}}_{1} is:$

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a positive real number

a negative real number

always zero

a purely imaginary numbe

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detailed solution

Correct option is C

|2+i|2=z1+z2+z32⇒3=z12+z22+z32+2Re⁡z1z¯2+z2z¯3+z3z¯1⇒ 3=1+1+1+2Re⁡z1z¯2+z2z¯3+z3z¯1⇒Re⁡z1z¯2+z2z¯3+z3z¯1=0

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If z1=z2=z3=1 and z1+z2+z3=2+i then the number z1z¯2+z2z¯3+z3z¯1 is:

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If $|z_{1}| = |z_{2}| = |z_{3}| = 1$ and $z_{1} + z_{2} + z_{3} = \sqrt{2} + i$ then the number $z_{1} {\bar{z}}_{2} + z_{2} {\bar{z}}_{3} + z_{3} {\bar{z}}_{1} is:$