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Complex Numbers

If |z1|=2, ...

Question

If $| z_{1} | = 2, | z_{2} | = 3, | z_{3} | = 4$ and $| z_{1} + z_{2} + z_{3} | = 5$ , then $| 4 z_{2} z_{3} + 9 z_{3} z_{1} + 16 z_{1} z_{2} | =$

A

20

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B

24

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C

48

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D

120

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Solution

The correct option is D 120
$| 4 z_{2} z_{3} + 9 z_{3} z_{1} + 16 z_{1} z_{2} |$
$= | z_{1}_{1} z_{2} z_{3} + z_{2}_{2} z_{3} z_{1} + z_{3}_{3} z_{1} z_{2} |$
$= | z_{1} z_{2} z_{3} | | ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ z_{1} + z_{2} + z_{3} |$
$= 2 \times 3 \times 4 \times 5$
$= 120$

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Similar questions

z_{1}, z_{2}, z_{3}

3

distinct complex numbers such that

\frac{3}{| z_{2} - z_{3} |} = \frac{4}{| z_{3} - z_{1} |} = \frac{5}{| z_{1} - z_{2} |}

, then the value of

\frac{9}{z_{2} - z_{3}} + \frac{16}{z_{3} - z_{1}} + \frac{25}{z_{1} - z_{2}}

z_{1}

z_{2}

z_{3}

are three distinct complex numbers such that

\frac{1}{∣ z_{1} - z_{2} ∣} = \frac{3}{∣ z_{2} - z_{3} ∣} = \frac{5}{∣ z_{1} - z_{3} ∣}

, then the value of

\frac{1}{z_{1} - z_{2}} + \frac{9}{z_{2} - z_{3}} + \frac{25}{z_{3} - z_{1}}

z_{1} + z_{2} + z_{3} = 0

| z_{2} - z_{3} |^{2} + | z_{3} - z_{1} |^{2} + | z_{1} - z_{2} |^{2}

| z_{1} | = | z_{2} | = | z_{3} | = 1

z_{1} + z_{2} + z_{3} + = \sqrt{2} + i

, then the complex number

z_{2}_{3} + z_{3} +_{1} + z_{1} +_{2}

| z_{1} | = 1, | z_{2} | = 2, | z_{3} | = 3

| 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} | i s

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