Q101 / 180 · COMEDK 2016 · Mathematics

The complex number $\frac{(-\sqrt{3}+3 i)(1-i)}{(3+\sqrt{3} i)(i)(\sqrt{3}+\sqrt{3} i)}$, when represented in the Argand diagram lies

  • A
    on the $X$-axis (Real axis)
  • Correct answer
    B
    on the $Y_{\text {-axis }}$ (Imaginary axis)
  • C
    in the first quadrant
  • D
    in the second quadrant

Solution

Let $\begin{aligned} &=\frac{(-\sqrt{3}+3 i)(1-i)}{(3+\sqrt{3} i)(i)(\sqrt{3}+\sqrt{3} i)} \\ &=\frac{\sqrt{3}(-1+\sqrt{3} i)(1-i)}{(\sqrt{3})^{2}(\sqrt{3}+i)(1+i)(i)} \\ &=\frac{(-1+\sqrt{3} i)(1-i)}{\sqrt{3}\left(\sqrt{3} i+i^{2}\right)(1+i)}=\frac{(-1+\sqrt{3 i})(1-i)}{\sqrt{3}(-1+\sqrt{3 i})(1+i)} \\ &=\frac{1-i}{(\sqrt{3}) 1+i} \times \frac{(1-i)}{(1-i)}=\frac{(1-i)^{2}}{(\sqrt{3})(1+1)} \\ &=\frac{1+i^{2}-2 i}{2 \sqrt{3}}=\frac{-2 i}{2 \sqrt{3}}=-\frac{i}{\sqrt{3}} \\ &=\text { Purely imaginary } \end{aligned}$

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

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