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QJEEM24Mad46b: JEE Main 2024 Mathematics - Class 11 JEEM24Mad46b Solution: The sum of all possible values of , for which is purely…

The sum of all possible values of θ∈[−π,2π]\theta \in[-\pi, 2 \pi] , for which 1+icos⁡θ1−2icos⁡θ\frac{1+i \cos \theta}{1-2 i \cos \theta} is purely imaginary, is equal to :

Multiple Choice Options

  • A 4π4 \pi
  • B 3π3 \pi
  • C 2π2 \pi
  • D 5π5 \pi

Step-by-Step Verified Solution

1+icos⁡θ1−2icos⁡θ\frac{1+i \cos \theta}{1-2 i \cos \theta} is purely imaginary

n=1+icos⁡θ1−2icos⁡θ×1+2icos⁡θ1+2icos⁡θ=1+3icos⁡θ−2cos⁡2θ1+4cos⁡2θn=\frac{1+i \cos \theta}{1-2 i \cos \theta} \times \frac{1+2 i \cos \theta}{1+2 i \cos \theta}=\frac{1+3 i \cos \theta-2 \cos ^2 \theta}{1+4 \cos ^2 \theta}

n=1−2cos⁡2θ1+4cos⁡2θ+i(3cos⁡θ1+4cos⁡2θ)n=\frac{1-2 \cos ^2 \theta}{1+4 \cos ^2 \theta}+i\left(\frac{3 \cos \theta}{1+4 \cos ^2 \theta}\right)

nn is purely imaginary

⇒1−2cos⁡2θ1+4cos⁡2θ=0\Rightarrow \frac{1-2 \cos ^2 \theta}{1+4 \cos ^2 \theta}=0

⇒cos⁡2θ=12\Rightarrow \cos ^2 \theta=\frac{1}{2}

⇒cos⁡θ=±12\Rightarrow \cos \theta= \pm \frac{1}{\sqrt{2}}

θ\theta can be π4,−π4,3π4,−3π4,5π4,7π4\frac{\pi}{4}, \frac{-\pi}{4}, \frac{3 \pi}{4}, \frac{-3 \pi}{4}, \frac{5 \pi}{4}, \frac{7 \pi}{4}

Sum of all possible values of θ=3π\theta=3 \pi

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