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JEE Main 20205 SeptemberEvening Shift
Exam
JEE Main
Year
2020
Date
5 September
Shift
Evening Shift
Subject
Mathematics
Chapter
Complex Number

JEE Main 2020 · 5 September · Evening Shift · Mathematics · Complex Number

The value of ( - 1 + i 3 1 - i ) ^ 30 is :

The value of${\left(\dfrac{- 1 + i\sqrt{3}}{1 - i}\right)}^{30} is :

Options

  1. A65{6}^{5}
  2. B215{2}^{15} i
  3. C−215- {2}^{15}
  4. D−215- {2}^{15}i

Correct answer

D. −215- {2}^{15}i

Step-by-step solution

(−1+3i1−i)30=(2cos⁡(2π3)+isin(2π3)2(cosπ4−isinπ4))30{\left(\dfrac{- 1 + \sqrt{3}i}{1 - i}\right)}^{30} = {\left(\dfrac{2\cos \left(\dfrac{2\pi}{3}\right) + isin\left(\dfrac{2\pi}{3}\right)}{\sqrt{2}\left(cos\dfrac{\pi}{4} - isin\dfrac{\pi}{4}\right)}\right)}^{30}

=230(cos⁡20π+isin20π)215(cos15π2−isin15π2)= \dfrac{{2}^{30} ( \cos 20\pi + isin20\pi )}{{2}^{15}\left(cos\dfrac{15\pi}{2} - isin\dfrac{15\pi}{2}\right)}

=215(1+0i)(0+i)=−215i= \dfrac{{2}^{15} ( 1 + 0i )}{( 0 + i )} = - {2}^{15}i

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