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If     i=\sqrt{-1}  then  4+5 \left ( \frac{-1}{2} + \frac{i \sqrt 3}{2} \right )^{334}- \:3 \left ( \frac{1}{2} + \frac{i \sqrt 3}{2} \right )^{365}     is equal to:

  • Option 1)

    1-i\sqrt 3

  • Option 2)

    -1+i\sqrt 3

  • Option 3)

    4\sqrt 3\: i

  • Option 4)

    -i\sqrt 3

 

Answers (1)

best_answer

As we learnt in 

Properties of Conjugate of Complex Number -

\bar{\bar{z}}=z

- wherein

\bar{z} denotes conjugate of z

 

 4+5\left ( \frac{-1}{2} +\frac{i\sqrt{3}}{2}\right )^{334}-3\left ( \frac{1}{2} +\frac{i\sqrt{3}}{2}\right )^{365}\\*\\*4+5\left ( \frac{-1}{2} +\frac{i\sqrt{3}}{2}\right )-\frac{3}{2}+\frac{3\sqrt{3i}}{2}\: \: \: \: \left [ w^{334}=w \: \: \: ,\: \: \: \\* \left ( w^{2} \right )^{365}=w \right ]\\*\\*4-4+4\sqrt{3}i=4\sqrt{3}i


Option 1)

1-i\sqrt 3

Incorrect

Option 2)

-1+i\sqrt 3

Incorrect

Option 3)

4\sqrt 3\: i

Correct

Option 4)

-i\sqrt 3

Incorrect

Posted by

divya.saini

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