MHT CET · Maths · Complex Number
If a complex number \(\mathrm{z}=\frac{4+3 i \sin \theta}{1-2 i \sin \theta}(\) where \(i=\sqrt{-1})\) is purely real, then the value of \(\theta\) is
- A \((\mathrm{n}+1) \frac{\pi}{2}, \mathrm{n} \in \mathbb{Z}\)
- B \((n-1) \frac{\pi}{2}, n \in \mathbb{Z}\)
- C \((2 n+1) \frac{\pi}{4}, n \in \mathbb{Z}\)
- D \(n \pi, n \in \mathbb{Z}\)
Answer & Solution
Correct Answer
(D) \(n \pi, n \in \mathbb{Z}\)
Step-by-step Solution
Detailed explanation
\(\mathrm{z}=\frac{4+3 i \sin \theta}{1-2 i \sin \theta} \times \frac{1+2 i \sin \theta}{1+2 i \sin \theta}\) \(\mathrm{z}=\frac{4+8i\sin\theta+3i\sin\theta-6\sin^2\theta}{1+4\sin^2\theta}\)
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