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MathematicsNumerical+3Question 5 of 54

JEE Advanced 2022 · Paper 1, Mathematics question 5

Let zˉ\bar z denote the complex conjugate of a complex number zz and let i=−1i = \sqrt{-1}. In the set of complex numbers, the number of distinct roots of the equation zˉ−z2=i(zˉ+z2)\bar z - z^2 = i(\bar z + z^2) is _____.

Answer and worked solution

Official answer: 4

Simplify. Rearranging gives zˉ(1−i)=z2(1+i)\bar z(1-i)=z^2(1+i), so
zˉ=1+i1−iz2=iz2.\bar z=\frac{1+i}{1-i}z^2=iz^2.

Roots.

  • z=0z=0 is a root.
  • Otherwise, taking moduli gives ∣z∣=∣z∣2|z|=|z|^2, so ∣z∣=1|z|=1 and zˉ=1z\bar z=\frac1z. Then 1z=iz2\frac1z=iz^2, i.e. z3=−iz^3=-i, which has 3 distinct roots.

That makes 4 distinct roots.

Topics:Algebra of complex numbersRoots of unityModulus and argument
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