Let w ⁡ = 3 + i 2 and P = w ⁡ n : n = 1, 2, 3, … .Further H 1 = z ∈ C : Re z > 1 2 and H 2 = z ∈ C : Re z < - 1 2 , where C is the set of all complex numbers. If z 1 ∈ P ∩ H 1 , z 2 ∈ P ∩ H 2 and O represents the origin, then ∠ z 1 O z 2 =

P = ω n = cos n π 6 + i sin n π 6 , H 1 = Rez > 1 / 2 z 1 = P ∩ H 1 = 3 + i 2 , 3 - i 2 z 2 = P ∩ H 2 = - 1 , - 3 + i 2 , - 3 - i 2 ∠ z 1 O z 2 = 2 π 3 , 5 π 6 , π

Home

Mathematics - Complex Number Question with Solution | TestHub

MathematicsComplex NumberGeometric ApplicationHard2 minPYQ_2013

MathematicsHardmultiple choice

Let $w = \frac{\sqrt{3} + i}{2}$ and $P = \{w^{n} : n = 1, 2, 3, \dots\}$ .Further $H_{1} = \{z \in C : Re z > \frac{1}{2}\}$ and $H_{2} = \{z \in C : Re z < \frac{- 1}{2}\}$ , 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 $∠ z_{1} O z_{2} =$

Question diagram: Let w ⁡ = 3 + i 2 and P = w ⁡ n : n = 1, 2, 3, … .Further H

Options:(select one or more)

Answer:

C, D

Solution:

$P = ω^{n} = cos \frac{n π}{6} + i sin \frac{n π}{6}, H_{1} = Rez > 1 / 2$
$z_{1} = P \cap H_{1} = \frac{\sqrt{3} + i}{2}, \frac{\sqrt{3} - i}{2}$
$z_{2} = P \cap H_{2} = - 1, \frac{- \sqrt{3} + i}{2}, \frac{- \sqrt{3} - i}{2}$
$∠ z_{1} O z_{2} = \frac{2 π}{3}, \frac{5 π}{6}, π$

Stream:JEE_ADVSubject:MathematicsTopic:Complex NumberSubtopic:Geometric Application

⏱ 2mℹ️ Source: PYQ_2013

Doubts & Discussion

Loading discussions...