Q199 / 270 · NDA 2016 (Phase 1) · Mathematics

Let $z_1, z_2$ and $z_3$ be non-zero complex numbers satisfying $\mathrm{z}^2=\mathrm{i} \overline{\mathrm{z}}$, where $\mathrm{i}=\sqrt{-1}$. Consider the following statements: 1. $z_1 z_2 z_3$ is purely imaginary. 2. $z_1 z_2+z_2 z_3+z_3 z_1$ is purely real. Which of the above statements is/are correct?

  • A
    1 only
  • B
    2 only
  • Correct answer
    C
    Both 1 and 2
  • D
    Neither 1 nor 2

Solution

$\begin{aligned} z_1 z_2 z_3 & =(-i)\left(\frac{\sqrt{3}+i}{2}\right)\left(\frac{-\sqrt{3}+i}{2}\right) \\ & =\frac{-i}{4}\left(i^2-(\sqrt{3})^2\right) \\ & =\frac{-i}{4}(-3-1)=i \end{aligned}$ Hence $z_1 z_2 z_3$ is purely imaginary. $\begin{aligned} z_1 z_2 & =-i\left(\frac{\sqrt{3}+i}{2}\right)=\frac{-\sqrt{3} i+1}{2} \\ z_2 z_3 & =\frac{(\sqrt{3}+i)(-\sqrt{3}+i)}{4} \\ & =\left(\frac{-3-\sqrt{3} i+\sqrt{3} i+i^2}{4}\right)=-1 \end{aligned}$ $\begin{aligned} & z_3 z_1=\frac{(-\sqrt{3}+i)(-i)}{2}=\frac{+\sqrt{3} i+1}{2} \\ & z_1 z_2+z_2 z_3+z_3 z_1=\left(\frac{-\sqrt{3} i+1}{2}\right)+(-1)+\left(\frac{\sqrt{3} i+1}{2}\right) \\ & =\left(\frac{-\sqrt{3} i+1+\sqrt{3} i+1}{2}\right)-1 \\ & =0 \in R \end{aligned}$ Hence $z_1 z_2+z_2 z_3+z_3 z_1=0$ is purely real. Hence both statements are correct.

Hence, the correct answer is option: 'C'.

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