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JEE Main 2013

JEE Main 2013 Mathematics question on Complex Numbers and Quadratic Equations

If zz is a complex number of unit modulus and argument θ\theta, then arg⁡(1+z1+zˉ)\arg\left(\dfrac{1+z}{1+\bar{z}}\right) is equal to
  1. A.
    −θ-\theta
  2. B.
    π2−θ\dfrac{\pi}{2} - \theta
  3. C.
    θ\theta
  4. D.
    π−θ\pi - \theta
Show answer and solution

Correct option: C

Solution

Since |z| = 1, we have z·z̄ = 1, so z̄ = 1/z.

Substitute: (1+z)/(1+z̄) = (1+z)/(1+1/z) = (1+z)/((z+1)/z) = z.

Therefore arg((1+z)/(1+z̄)) = arg(z) = θ.

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