Q45 / 120 · VITEEE 2008 · Mathematics

Argument of the complex number $\left(\frac{-1-3 i}{2+i}\right)$ is

  • A
    $45^{\circ}$
  • B
    $135^{\circ}$
  • Correct answer
    C
    $225^{\circ}$
  • D
    $240^{\circ}$

Solution

We have, $\begin{array}{l} \frac{-1-3 \mathrm{i}}{2+\mathrm{i}}=\frac{-1-3 \mathrm{i}}{2+\mathrm{i}} \times \frac{2-\mathrm{i}}{2-\mathrm{i}}=\frac{-2-5 \mathrm{i}+3 \mathrm{i}^{2}}{4-\mathrm{i}^{2}} \\ =-1-\mathrm{i} \end{array}$ Now, let us put $-1=r \cos \theta,-1=r \sin \theta$ Squaring and adding, $r^{2}=2$ i.e., $r=\sqrt{2}$ So that $\cos \theta=\frac{-1}{\sqrt{2}}, \sin \theta=\frac{-1}{\sqrt{2}}$ Therefore, $\theta=225^{\circ}$ Thus, argument is $225^{\circ}$.

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

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