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JEE MATHEMATICS MCQ: COMPLEX NUMBERS

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Let SS be the set of all (α,β),π<α,β<2π(\alpha, \beta), \pi<\alpha, \beta<2 \pi, for which the complex number 1−isin⁡α1+2isin⁡α\frac{1-i \sin \alpha}{1+2 i \sin \alpha} is purely imaginary and 1+icos⁡β1−2icos⁡β\frac{1+i \cos \beta}{1-2 i \cos \beta} is purely real, Let Zαβ=sin⁡2α+iZ_{\alpha \beta}=\sin 2 \alpha+i cos⁡2β,(α,β)∈S\cos 2 \beta,(\alpha, \beta) \in S. Then ∑(α, β)∈S(iZαβ+1iZˉαβ)\sum\limits_{\left( {\alpha ,\,\beta } \right) \in S} {\left( {i{Z_{\alpha \beta }} + \frac{1}{{i{{\bar Z}_{\alpha \beta }}}}} \right)} is equal to
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Correct option: C
1
Option C (1) is correct because after finding the values of alpha and beta that satisfy the given conditions, substituting them into the expression and simplifying yields 1.

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