MHT CET · Maths · Complex Number
\(A\) value of \(\theta\), for which \(\frac{2+3 i \sin \theta}{1-2 \sin \theta}, \mathrm{i}=\sqrt{-1}\) is purely imaginary, is
- A \(\frac{\pi}{6}\)
- B \(\frac{\pi}{3}\)
- C \(\sin ^{-1}\left(\frac{1}{\sqrt{3}}\right)\)
- D \(\sin ^{-1}(\sqrt{3})\)
Answer & Solution
Correct Answer
(C) \(\sin ^{-1}\left(\frac{1}{\sqrt{3}}\right)\)
Step-by-step Solution
Detailed explanation
\(\mathrm{z}=\frac{2+3 i \sin \theta}{1-2 i \sin \theta} \times \frac{1+2 i \sin \theta}{1+2 i \sin \theta}=\frac{\left(2-6 \sin ^2 \theta\right)+\mathrm{i}(7 \sin \theta)}{1+4 \sin ^2 \theta}\)
for \(z\) to be purely imaginary \(\operatorname{Re}(z)=0\)
\(\Rightarrow 6 \sin ^2 \theta=2\)
\(\Rightarrow \theta=\sin ^{-1}\left(\frac{1}{\sqrt{3}}\right)\)
for \(z\) to be purely imaginary \(\operatorname{Re}(z)=0\)
\(\Rightarrow 6 \sin ^2 \theta=2\)
\(\Rightarrow \theta=\sin ^{-1}\left(\frac{1}{\sqrt{3}}\right)\)
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