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  • Tell me? - Complex numbers and quadratic equations - JEE Main-15

Arg \left ( \frac{2i}{\sqrt{3}-i} \right ) equals

  • Option 1)

    \frac{\pi }{6}

  • Option 2)

    \frac{\pi }{4}

  • Option 3)

    \frac{\pi }{3}

  • Option 4)

    \frac{2\pi }{3}

 

Answers (1)

\frac{2i}{\sqrt{3}-i}=\frac{2i}{\sqrt{3}-i}\times \frac{\sqrt{3}+i}{\sqrt{3}+i}=\frac{2\sqrt{3}i-2}{4}

\Rightarrow \: \frac{2i}{\sqrt{3}-i}=\frac{-1}{2}+\frac{i\sqrt{3}}{2}

\because it lies in 2nd quadrant so 

argument= \pi -\tan ^{-1}\left | \frac{\frac{\sqrt{3}}{2}}{\frac{-1}{2}} \right |=\pi -\frac{\pi }{3}=\frac{2\pi }{3}

\therefore Option (D)

 

Definition of Argument/Amplitude of z in Complex Numbers -

\theta =tan^{-1}|\frac{y}{x}|, z\neq 0

\boldsymbol{\theta,\pi-\theta,-\pi+\theta,-\theta} are Principal Argument if z lies in first, second, third or fourth quadrant respectively.

- wherein

 

 


Option 1)

\frac{\pi }{6}

This is incorrect

Option 2)

\frac{\pi }{4}

This is incorrect

Option 3)

\frac{\pi }{3}

This is incorrect

Option 4)

\frac{2\pi }{3}

This is correct

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subam

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