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JEE MainMathematicsModerateComplex Numbers

Let
A={θ∈(−π2,π):3+2isinθ1−2isinθ ispurely imaginary }A = \left\{ \theta \in \left( - \frac{\pi}{2},\pi \right):\frac{3 + 2isin\theta}{1 - 2isin\theta}\ ispurely\ imaginary\ \right\}.
Then the sum of the elements in A is

Show answer and solution
 Let z=3+2isinθ1−2isinθθ∈(−π2,π)\ Let\ z = \frac{3 + 2isin\theta}{1 - 2isin\theta}\theta \in \left( - \frac{\pi}{2},\pi \right)
for z to be purely imaginary
z+z¯=0z + \overset{¯}{z} = 0
⇒3+2isinθ1−2isinθ+3−2isinθ1+2isinθ=0\Rightarrow \frac{3 + 2isin\theta}{1 - 2isin\theta} + \frac{3 - 2isin\theta}{1 + 2isin\theta} = 0
⇒3+6isinθ+2isinθ−4sin⁡2θ+3−6isinθ−2isinθ−4sin⁡2θ1+4sin⁡2θ\Rightarrow \frac{3 + 6isin\theta + 2isin\theta - 4\sin^{2}\theta + 3 - 6isin\theta - 2isin\theta - 4\sin^{2}\theta}{1 + 4\sin^{2}\theta}
⇒\Rightarrow 3-4sin^2θ=0 ⇒\Rightarrow sin^2θ = 3/4
⇒\Rightarrowθ=nπ±π3\pi \pm \frac{\pi}{3}
Given θ ∈(−π2,π)\in \left( \frac{- \pi}{2},\pi \right)
Hence θ = −π3, π3, 2π3\frac{- \pi}{3},\ \frac{\pi}{3},\ \frac{2\pi}{3}
Sum of all possible values of θ = 2π3\frac{2\pi}{3}

Source: Grass-G / M-28 Mathematics Paper FST 16 Jan.docx

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