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  • The real value of θ for which the expression is a real number is: A. B. C. D. none of these

The real value of θ for which the expression  \frac{1+i \cos \theta}{1-2 i \cos \theta} is a real number is:
A. n\pi+\frac{\pi}{4} 

B.  n\pi+(-1)^n\frac{\pi}{4}
C. 2n\pi+\frac{\pi}{2}
D. none of these

Answers (1)

Let \: \: z=\frac{1+i \cos \theta}{1-2 i \cos \theta} =\frac{1+i \cos \theta}{1-2 i \cos \theta} * \frac{1+2 i \cos \theta}{1+2 i \cos \theta}

On solving we get

\frac{1+3 i \cos \theta-2 \cos ^{2} \theta}{1-4 i^{2} \cos ^{2} \theta} \\=\frac{1+3 i \cos \theta-2 \cos ^{2} \theta}{1+4 \cos ^{2} \theta}

=\frac{\left(1-2 \cos ^{2} \theta\right)+3 i \cos \theta}{1+4 \cos ^{2} \theta}

If z is real number then

\frac{3 \cos \theta}{1+4 \cos ^{2} \theta}=0

\\3 \cos \theta=0 \\\cos \theta=0

\theta=\frac{(2 n+1) \pi}{2}, n \in N

Posted by

infoexpert21

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