z1,z2,z3are three complex numbers such that z1=1, z2=2,z3=3,,and z1+z2+z3=2,then 3z1z2+z2z3+2z3z1=−−−−

3z1z2+z2z3+2z3z1=z32z1z2+z12z2z3+z22z3z1=z3z3¯z1z2+z1z1¯z2z3+z2z2¯z3z1=z1z2z3z1+z2+z3=1232=26

z1,z2,z3are three complex numbers such that z1=1, z2=2,z3=3,,and z1+z2+z3=2,then 3z1z2+z2z3+2z3z1=−−−−

3z1z2+z2z3+2z3z1=z32z1z2+z12z2z3+z22z3z1=z3z3¯z1z2+z1z1¯z2z3+z2z2¯z3z1=z1z2z3z1+z2+z3=1232=26

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$z_{1}, z_{2}, z_{3}$ are three complex numbers such that $|z_{1}| = 1,$ $|z_{2}| = \sqrt{2},$ $|z_{3}| = \sqrt{3},$ ,and $|z_{1} + z_{2} + z_{3}| = 2,$ then $|3 z_{1} z_{2} + z_{2} z_{3} + 2 z_{3} z_{1}| = - - - -$

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$2 \sqrt{2}$

$2 \sqrt{3}$

$\sqrt{6}$

$2 \sqrt{6}$

answer is D.

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$|3 z_{1} z_{2} + z_{2} z_{3} + 2 z_{3} z_{1}| = |{|z_{3}|}^{2} z_{1} z_{2} + {|z_{1}|}^{2} z_{2} z_{3} + {|z_{2}|}^{2} z_{3} z_{1}|$

$= |z_{3} \bar{z_{3}} z_{1} z_{2} + z_{1} \bar{z_{1}} z_{2} z_{3} + z_{2} \bar{z_{2}} z_{3} z_{1}|$

$= |z_{1}| |z_{2}| |z_{3}| |z_{1} + z_{2} + z_{3}| = (1) (\sqrt{2}) (\sqrt{3}) (2) = 2 \sqrt{6}$

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z1,z2,z3are three complex numbers such that z1=1, z2=2,z3=3,,and z1+z2+z3=2,then 3z1z2+z2z3+2z3z1=−−−−

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$z_{1}, z_{2}, z_{3}$ are three complex numbers such that $|z_{1}| = 1,$ $|z_{2}| = \sqrt{2},$ $|z_{3}| = \sqrt{3},$ ,and $|z_{1} + z_{2} + z_{3}| = 2,$ then $|3 z_{1} z_{2} + z_{2} z_{3} + 2 z_{3} z_{1}| = - - - -$