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  • Try this! - Complex numbers and quadratic equations - JEE Main-11

Arg \left ( i^{18}+\frac{1}{i^{25}} \right )  equals

  • Option 1)

    \frac{-\pi }{4}

  • Option 2)

    \frac{-\pi }{2}

  • Option 3)

    \frac{-3\pi }{4}

  • Option 4)

    \frac{\pi }{4}

 

Answers (1)

best_answer

 i^{18}+\frac{1}{i^{25}} =\left ( i^{4} \right )\cdot i^{2}+\left ( \frac{1}{i^{4}} \right )^{6}\cdot \frac{1}{i}=i^{2}+\frac{1}{i}= -1-i

\therefore  Arg \left ( -1-i \right )=-\pi +\tan ^{-1}\left | \frac{-1}{-1} \right |=-\pi +\frac{\pi }{4}=\frac{-3\pi }{4}

\therefore Option (C)

 

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 }{4}

This is incorrect

Option 2)

\frac{-\pi }{2}

This is incorrect

Option 3)

\frac{-3\pi }{4}

This is correct

Option 4)

\frac{\pi }{4}

This is incorrect

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prateek

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