Let A = \ θ ∈ (- (π)/(2), π): (3 + 2isinθ)/(1 - 2isinθ)ispurelyimaginary\ . Then the sum of the elements in A is:

Correct answer: C. (2π)/(3) Given complex number is z = (3 + 2isinθ)/(1 - 2isinθ) ⇒ z = ( 3 + 2isinθ ) ( 1 + 2isinθ ) ( 1 + 4 ^ 2 θ ) Since given complex number is purely imaginary ⇒ Re ( z ) = 0 ⇒ 3 - 4 ^ 2 θ = 0 ⇒ sinθ = ± 3 2 ⇒ θ = (- π)/(3), (π)/(3), (2π)/(3) ⇒ sum of all possible values of θ = (2π)/(3) .

JEE Main 20199 JanuaryMorning Shift

Exam: JEE Main
Year: 2019
Date: 9 January
Shift: Morning Shift
Subject: Mathematics
Chapter: Complex Number

JEE Main 2019 · 9 January · Morning Shift · Mathematics · Complex Number

Let A = \ θ ∈ (- (π)/(2), π): (3 + 2isinθ)/(1 -...

Let

A = \left\{\theta \in \left(- \dfrac{\pi}{2}, \pi\right): \dfrac{3 + 2isin\theta}{1 - 2isin\theta}ispurelyimaginary\right\} .

Then the sum of the elements in

A

is:

Options

A $\dfrac{5\pi}{6}$
B $\pi$
C $\dfrac{2\pi}{3}$
D $\dfrac{3\pi}{4}$

Correct answer

C. $\dfrac{2\pi}{3}$

Step-by-step solution

Given complex number is

z = \dfrac{3 + 2isin\theta}{1 - 2isin\theta}

⇒ z = \dfrac{( 3 + 2isin\theta ) ( 1 + 2isin\theta )}{( 1 + 4\sin^{2}\theta )}

Since given complex number is purely imaginary

⇒ Re ( z ) = 0

⇒ 3 - 4\sin^{2}\theta = 0

⇒ sin\theta = \pm \dfrac{\sqrt{3}}{2}

⇒ \theta = \dfrac{- \pi}{3}, \dfrac{\pi}{3}, \dfrac{2\pi}{3}

⇒

sum of all possible values of

\theta = \dfrac{2\pi}{3}

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