Find real θ such that 1−2 i cosθ 1+i cosθ is purely imaginary

Given, 1−2icosθ 1+icosθ =1+4cos2θ (1+icosθ )(1+2icosθ ) =1+4cos2θ 1+2icosθ +icosθ −2cos2θ =1+4cos2θ 1−2cos2θ +3icosθ z is purely imaginary when 1−2cos2θ =0⇒cosθ =±2 1 θ=nπ±4π Ans: D

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Find real such that {1 + i }{1 - 2 i } is purely imagi

Question

Find real $θ$ such that $\frac{1 + i cos θ}{1 - 2 i cos θ}$ is purely imaginary

nπ \pm \frac{π}{2}

nπ \pm \frac{π}{6}

nπ \pm \frac{π}{3}

nπ \pm \frac{π}{4}

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Jack Discussed

Find real

θ

such that

\frac{1 + i cos θ}{1 - 2 i cos θ}

is purely imaginary

8 mins ago

Discuss this question LIVE

8 mins ago

Text solutionVerified

Given,

\frac{1 + i cos θ}{1 - 2 i cos θ} = \frac{( 1 + i cos θ ) ( 1 + 2 i cos θ )}{1 + 4 cos ^{2} θ}

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

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

z

is purely imaginary when

1 - 2 cos^{2} θ = 0

\Rightarrow cos θ = \pm \frac{1}{2}

θ = nπ \pm \frac{π}{4}

Ans: D

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Question Text	Find real $θ$ such that $\frac{1 + i cos θ}{1 - 2 i cos θ}$ is purely imaginary
Answer Type	Text solution:1
Upvotes	150

Find real θ such that 1−2 i cosθ 1+i cosθ​ is purely imaginary

Text solutionVerified

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Find real $θ$ such that $\frac{1 + i cos θ}{1 - 2 i cos θ}$ is purely imaginary