If z 1- z 2- z 3 are complex numbers such that - z 1- z 2- z 3-1- z 1-1- z 2-1- z 3-1- then - z 1- z 2- z 3- isA- equal to 1B- less than 1C- greater than 3D- equal to 3

The correct option is A equal to 1|z_1|=1⇒ z_1z̅_̅1̅=1⇒1/z_1=z̅_̅1̅ Similarly, 1/z_2=z̅_̅2̅ and 1/z_3=z̅_̅3̅ Now, |z_1+z_2+z_3|=|z_1+z_2+z_3| =|z̅_̅1̅+z̅_̅2̅+z ...

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Byju's Answer

Standard XI

Mathematics

Properties of Modulus

If z 1, z 2, ...

Question

If $z_{1}, z_{2}, z_{3}$ are complex numbers such that $| z_{1} | = | z_{2} | = | z_{3} | = ∣ ∣ \frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}} ∣ ∣ = 1$ , then $| z_{1} + z_{2} + z_{3} |$ is

equal to 1

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less than 1

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greater than 3

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equal to 3

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Solution

The correct option is A equal to 1
$| z_{1} | = 1 \Rightarrow z_{1}_{1} = 1 \Rightarrow \frac{1}{z_{1}} =_{1}$
Similarly, $\frac{1}{z_{2}} =_{2}$ and $\frac{1}{z_{3}} =_{3}$
Now, $| z_{1} + z_{2} + z_{3} | = | ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ z_{1} + z_{2} + z_{3} |$
= $|_{1} +_{2} +_{3} |$
= $∣ ∣ \frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}} ∣ ∣ = 1$

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Similar questions

If $z_{1}, z_{2} a n d z_{3}$ are complex numbers such that

| $z_{1}$ |=| $z_{2}$ |=| $z_{3}$ |= $∣ ∣ \frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}} ∣ ∣ = 1$ ,

then | $z_{1}$ + $z_{2}$ + $z_{3}$ |

Q. If

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

are imaginary numbers such that

| z_{1} | = | z_{2} | = | z_{3} | = ∣ ∣ ∣ \frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}} ∣ ∣ ∣ = 1

then

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

if $z_{1}, z_{2} a n d z_{3}$ are complex numbers such that

| $z_{1}$ |=| $z_{2}$ |=| $z_{3}$ |= $∣ ∣ \frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}} ∣ ∣ = 1$ ,

then | $z_{1}$ + $z_{2}$ + $z_{3}$ |

If $Z_{1}, Z_{2}, Z_{3}$ are complex numbers such that

$| Z_{1} | = | Z_{2} | = | Z_{3} | = | \frac{1}{Z_{1}} + \frac{1}{Z_{2}} + \frac{1}{Z_{3}} | = 1$ then $| Z_{1} + | Z_{2} + Z_{3} |$ is

Q. If

z_{1}, z_{2}

and

z_{3}

are complex numbers such that

| z_{1} | = | z_{2} | = | z_{3} | = ∣ ∣ ∣ \frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}} ∣ ∣ ∣ = 1

then

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

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

The correct option is A equal to 1|z1|=1⇒z1¯z1=1⇒1z1=¯z1 Similarly, 1z2=¯z2 and 1z3=¯z3 Now, |z1+z2+z3|=|¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯z1+z2+z3| =|¯z1+¯z2+¯z3| =∣∣1z1+1z2+1z3∣∣=1

If $z_{1}, z_{2}, z_{3}$ are complex numbers such that $| z_{1} | = | z_{2} | = | z_{3} | = ∣ ∣ \frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}} ∣ ∣ = 1$ , then $| z_{1} + z_{2} + z_{3} |$ is