JEE Mains · Maths · STD 11 - 4.1 complex nubers
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 \(\sum_{(\alpha, \beta) \in s }\left(i Z_{\alpha \beta}+\frac{1}{i \bar{Z}_{\alpha \beta}}\right)\) is equal to.
- A \(3\)
- B \(3\,i\)
- C \(1\)
- D \(2-i\)
Answer & Solution
Correct Answer
(C) \(1\)
Step-by-step Solution
Detailed explanation
\(\pi<\alpha, \beta<2 \pi\) \(\frac{1-i \sin \alpha}{1+i(2 \sin \alpha)}=\text { Purely imaginary }\) \(\Rightarrow \frac{(1-i \sin \alpha)(1-i(2 \sin \alpha))}{1+4 \sin ^{2} \alpha}=\text { Purely imaginary }\) \(\frac{1-2 \sin ^{2} \alpha}{1+4 \sin ^{2} \alpha}=0\)…
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