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If $| Z_{1} | = 1, | Z_{2} | = 2, | Z_{3} | = 3$ and $| 9 Z_{1} Z_{2} + 4 Z_{1} Z_{3} + Z_{2} Z_{3} | = 12$ then the value of $| Z_{1} + Z_{2} + Z_{3} |$ is

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Detailed Solution

$| 9 z_{1} z_{2} + 4 z_{1} z_{3} + z_{2} z_{3} | = 12$

$| z_{3} {\bar{z}}_{3} z_{1} z_{2} + z_{2} {\bar{z}}_{2} z_{1} z_{3} + z_{1} \bar{z_{1}} z_{2} z_{3} | = 12$

$| z_{1} z_{2} z_{3} ({\bar{z}}_{3} + {\bar{z}}_{2} + {\bar{z}}_{1}) | = 12$

$| z_{1} | | z_{2} | | z_{3} | | z_{1} + z_{2} + z_{3} | = 12 \Rightarrow | z_{1} + z_{2} + z_{3} | = 2$

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