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NDA 2022
Exam
NDA
Year
2022
Subject
Mathematics
Chapter
Complex Number

NDA 2022 · Mathematics · Complex Number

Paragraph: Let z=(1+i θ)/(1-i θ) where i= -1 Question: What is...

Paragraph: Let z=1+isin⁡θ1−isin⁡θz=\frac{1+i \sin \theta}{1-i \sin \theta} where i=−1i=\sqrt{-1}
Question: What is angle θ\theta such that zz is purely imaginary? where nn is an integer

Options

  1. Anπ2\frac{n \pi}{2}
  2. B(2n+1)π2\frac{(2 n+1) \pi}{2}
  3. Cnπn \pi
  4. D2nπ2 n \pi

Correct answer

B. (2n+1)π2\frac{(2 n+1) \pi}{2}

Step-by-step solution

If zz is purely imaginary. ∴cos⁡2θ1+sin⁡2θ=0⇒cos⁡2θ=0⇒cos⁡θ=0⇒θ=(2n+1)π2\begin{array}{ll} \therefore \frac{\cos ^2 \theta}{1+\sin ^2 \theta}=0 & \Rightarrow \cos ^2 \theta=0 \\ \Rightarrow \cos \theta=0 & \Rightarrow \theta=(2 n+1) \frac{\pi}{2} \end{array} Where nn is an integer.
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