Complex Numbers
Algebra of Complex Numbers
JEE Main 2020
Grade 11
Question:
The real part of $\frac{1}{2 - e^{i\theta}}$ is:
(1) $\frac{2 - \cos \theta}{5 - 4 \cos \theta}$
(2) $\frac{2 + \cos \theta}{5 + 4 \cos \theta}$
(3) $\frac{1}{2 - \cos \theta}$
(4) $\frac{\cos \theta}{5 - 4 \cos \theta}$
Step-by-Step Solution
Key Concept: Substitute $e^{i\theta} = \cos \theta + i \sin \theta$. Multiply numerator and denominator by the conjugate $(2 - \cos \theta) + i \sin \theta$ and read off the real part.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)