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

The correct options are C 5π/6 D 2π/3 w=√(3)+i2=e^iπ/6, So #160; w^n=e^i(nπ/6) Now,for z_1, cosnπ/6 gt;1/2 and for z_2, cosnπ ...

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Byju's Answer

Standard XII

Mathematics

AM,GM,HM Inequality

Let w=√3+i/...

Question

Let $w = \frac{\sqrt{3} + i}{2}$ and $P = {w^{n} : n = 1, 2, 3, \dots \dots}$ . 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 options are
C $\frac{5 π}{6}$
D $\frac{2 π}{3}$

$w = \frac{\sqrt{3} + i}{2} = e^{\frac{i π}{6}},$

So $w^{n} = e^{i (\frac{n π}{6})}$

Now,

for $z_{1}, cos \frac{n π}{6} > \frac{1}{2}$ and for $z_{2}, cos \frac{n π}{6} < - \frac{1}{2}$

Possible position $o f z_{1}$ are $A_{1}, A_{2}, A_{3}$ whereas
$o f z_{2}$ are $B_{1}, B_{2}, B_{3}$ (as shown in the figure)

So, possible value of $∠ z_{1} O z_{2}$
according to the given options is $\frac{2 π}{3}$ or $\frac{5 π}{6}$

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Similar questions

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

The correct options are C 5π6 D 2π3w=√3+i2=eiπ6,So wn=ei(nπ6)Now,for z1, cosnπ6>12 and for z2, cosnπ6<−12Possible position of z1 are A1, A2, A3 whereas ofz2 are B1, B2, B3 (as shown in the figure) So, possible value of ∠z1Oz2 according to the given options is 2π3 or5π6