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Question
The real value of θ for which the expression `(1 + i cos theta)/(1 - 2i cos theta)` is a real number is ______.
Options
`npi + pi/4`
`npi + (-1)n pi/4`
`2npi +- pi/2`
None of these
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Solution
The real value of θ for which the expression `(1 + i cos theta)/(1 - 2i cos theta)` is a real number is `underlinebb(2npi +- pi/2)`.
Explanation:
Let z = `(1 + i cos theta)/(1 - 2i cos theta)`
= `(1 + i cos theta)/(1 - 2i cos theta) xx (1 + 2i cos theta)/(1 + 2i cos theta)`
= `(1 + 2i cos theta + i cos theta + 2i^2 cos^2 theta)/(1 - 4i^2 cos^2 theta)`
= `(1 + 3i cos theta - 2 cos^2 theta)/(1 + 4 cos^2 theta)`
= `(1 - 2 cos^2 theta)/(1 + 4 cos^2 theta) + (3 cos theta)/(1 + 4 cos^2 theta)i`
If z is a real number, then
`(3 cos theta)/(1 + 4cos^2 theta)` = 0
⇒ 3cosθ = 0
⇒ cosθ = 0
∴ θ = `2npi +- pi/2`, n ∈ N.
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