Let S be the set of all (α, β), π<α, β<2 π, for which the complex number (1-i sin α)/(1+2 i sin α) is purely imaginary and (1+i cos β)/(1-2 i cos β) is purely real. Let Z α β = sin 2 α+i cos 2 β,(α, β) S. Then Σ (α, β) S (i Z α β +1/(i Z α β)) is equal to:

Complex Numbers and Quadratic Equations: JEE Main 2022 Mathematics question

JEE Main 2022 · 27 Jul, Evening shift · Q3159

JEE Main 202227 Jul, Evening shiftMathematicsHardComplex Numbers and Quadratic Equations264× since 2002

Let S be the set of all

(\alpha, \beta), \pi<\alpha, \beta<2 \pi

, for which the complex number

\frac{1-i \sin \alpha}{1+2 i \sin \alpha}

is purely imaginary and

\frac{1+i \cos \beta}{1-2 i \cos \beta}

is purely real. Let

Z_{\alpha \beta}=\sin 2 \alpha+i \cos 2 \beta,(\alpha, \beta) \in S

. Then

\sum\limits_{(\alpha, \beta) \in S}\left(i Z_{\alpha \beta}+\frac{1}{i \bar{Z}_{\alpha \beta}}\right)

is equal to :