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What is the argument of the complex number \(\rm \frac{1 - i\sqrt3}{1 + i\sqrt3}\) where \({\rm{i}} = √ { - 1} \) ?
1. 60° 
2. 120° 
3. 240°  
4. 210° 

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Correct Answer - Option 3 : 240°  

Concept:

The argument of a complex number z = x + iy

arg(z) = tan-1\(\rm \left(y\over x\right)\)

The angle is according to the sign of the y and x

  • Both positive then angle ∈ [0°, 90°]
  • Negative x and positive y then angle ∈ [90°, 180°]
  • Negative x and negative y then angle ∈ [180°, 270°]
  • Positive x and negative y then angle ∈ [270°, 360°]

Calculation:

Given that:

\(\rm \frac{1 - i√3}{1 + i√3}\)

Multiply numerator and denominator both by (1 - i√3), we get

\(\rm (\frac{1 - i√3}{1 + i√3}) \times (\frac {1 - i√3}{1 - i √3})\) = \(\rm \frac{(1 - i√3)^2}{(1)^2 - (i√3)^2}\) = \(\rm \frac{[1 + (i√3)^2 - 2i√3]}{1 + 3}\) = \(\rm \frac{(- 2 - 2i√3]}{4}\) = \(\rm \frac{-1}{2} - \frac{√3}{2}i\)

For a complex number, Z = x + yi

Argument (θ) = tan-1(y/x)

Here, x = \(\rm -1\over2\) and y = \(\rm -{√3\over2}i\)

Argument (θ) = tan-1(y/x) = tan-1\(\rm \frac {(\frac{√3}{2})}{(\frac{1}{2})}\) = tan-1\(\rm (√ 3)\)

Here θ = 60° But x and y is negative so the angle lies in 3rd quardrant 

argument (θ) = 180° + 60° = 240°  

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