Let - -3-i2 and P -n - n - 1- 2- 3- Further H1-z - - Re z 12 and H2-z - - Re z- -12- where - is the set of all complex numbers- If z1- P - H1- z2 - P - H2 and O represents the origin- then - z1Oz2 -

Ω nbsp; =√(3)+i2=e^iπ/6 P ={ω^n : n = 1, 2, 3,⋯} ⇒ P=e^i(nπ/6), |P|=1 ⇒ P nbsp;lies on a unit circle centred at origin lying at a difference of angle of π6 ...

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Byju's Answer

Standard XII

Mathematics

Method of Intervals

Let ω =√3+i2 ...

Question

Let $ω = \frac{\sqrt{3} + i}{2}$ and $P = {ω^{n} : n = 1, 2, 3, \dots} .$ Further $H_{1} = {z \in C : R e z > \frac{1}{2}}$ and $H_{2} = {z \in C : R e z < - \frac{1}{2}},$ where $C$ is the set of all complex numbers. If $z_{1} \in P \cap H_{1},$ $z_{2} \in P \cap H_{2}$ and $O$ represents the origin, then $∠ z_{1} O z_{2} =$

\frac{π}{2}

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\frac{π}{6}

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\frac{2 π}{3}

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\frac{5 π}{6}

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Solution

The correct option is D $\frac{5 π}{6}$
$ω = \frac{\sqrt{3} + i}{2} = e^{i π / 6}$
$P = {ω^{n} : n = 1, 2, 3, \dots}$
$\Rightarrow P = e^{i (n π / 6)}, | P | = 1$
$\Rightarrow P$ lies on a unit circle centred at origin lying at a difference of angle of $\frac{π}{6} .$

$H_{1} = {z \in C : R e z > \frac{1}{2}}$ and
$H_{2} = {z \in C : R e z < - \frac{1}{2}}$
Now, for $z_{1}, cos (\frac{n π}{6}) > \frac{1}{2}$
So, possible positions of $z_{1}$ are $A, L$ and $K .$
Similarly, for $z_{2}, cos (\frac{n π}{6}) < - \frac{1}{2}$
So, possible positions of $z_{2}$ are $E, F$ and $G .$

$∴ ∠ z_{1} O z_{2}$ can be $\frac{2 π}{3}, \frac{5 π}{6}, π, \frac{7 π}{6}, \frac{4 π}{3} .$

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Similar questions

Q. Let

w = \frac{\sqrt{3} + i}{2}

and

P = {w^{n} : n = 1, 2, 3, \dots \dots}

. Further

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and

H_{2} = {z \in C : R e z < \frac{- 1}{2}}

, where

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z_{1} \in P \cap H_{1}, z_{2} \in P \cap H_{2}

and

O

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∠ z_{1} O z_{2} =

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S_{1} = {z \in C : | z - 3 - 2 i |^{2} = 8},

S_{2} = {z \in C : R e (z) \geq 5}

and

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S = S_{1} \cap S_{2} \cap S_{3},

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S_{1} = {z \in C : | z | < 4},

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Let $A = {z_{1} : z_{1}^{16} = 1, z_{1} ϵ C}, B = {z_{2} : z_{2}^{72} = 1, z_{2} ϵ C} a n d P = {z_{1} z_{2} : z_{1} ϵ A, z_{2} ϵ B}$ are three sets of complex roots of unity (where C denotes set of complex numbers), then

Q. Let

S = S_{1} \cap S_{2} \cap S_{3},

where

S_{1} = {z \in C : | z | < 4},

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Let ω=√3+i2 and P={ωn:n=1,2,3,⋯}. Further H1={z∈C:Re z>12} and H2={z∈C:Re z<−12}, where C is the set of all complex numbers. If z1∈P∩H1, z2∈P∩H2 and O represents the origin, then ∠z1Oz2=