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Complex Numbers question

2022 · 27 Jul · Shift 2 · Q23
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  5. /2022 · 27 Jul · Shift 2 · Q23

Complex Numbers question

2022 · 27 Jul · Shift 2 · Q23

JEE MainMathematicsComplex NumbersMCQ+4 / −1
Let S be the set of all (α,β),π<α,β<2π(\alpha, \beta), \pi\lt \alpha, \beta\lt 2 \pi(α,β),π<α,β<2π, for which the complex number 1−isin⁡α1+2isin⁡α\frac{1-i \sin \alpha}{1+2 i \sin \alpha}1+2isinα1−isinα​ is purely imaginary and 1+icos⁡β1−2icos⁡β\frac{1+i \cos \beta}{1-2 i \cos \beta}1−2icosβ1+icosβ​ is purely real. Let Zαβ=sin⁡2α+icos⁡2β,(α,β)∈SZ_{\alpha \beta}=\sin 2 \alpha+i \cos 2 \beta,(\alpha, \beta) \in SZαβ​=sin2α+icos2β,(α,β)∈S. Then ∑(α,β)∈S(iZαβ+1iZˉαβ)\sum\limits_{(\alpha, \beta) \in S}\left(i Z_{\alpha \beta}+\frac{1}{i \bar{Z}_{\alpha \beta}}\right)(α,β)∈S∑​(iZαβ​+iZˉαβ​1​) is equal to :
  1. A
    3
  2. B
    3 i
  3. C
    1
  4. D
    2 −-− i
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