If z=\frac{\sqrt{3}}{2}+\frac{i}{2}\left ( i=\sqrt{-1} \right ), then \left ( 1+iz+z^{5}+iz^{8} \right )^{9} is equal to : 

 

 

  • Option 1)

    0

  • Option 2)

    1

  • Option 3)

    \left ( -1+2i \right )^{9}

     

  • Option 4)

    -1

 

Answers (1)

z=\frac{\sqrt{3}}{2}+\frac{i}{2}\Rightarrow iz=\frac{\sqrt{3}}{2}i-\frac{1}{2}=w   (cube root of unity)

\therefore \left ( 1+iz+z^{5}+iz^{8} \right )^{9}

\Rightarrow \left ( 1+w+\frac{w^{5}}{i^{5}}+i\times \frac{w^{8}}{i^{8}} \right )^{9}=\left ( 1+w+(-iw^{2})+iw^{2} \right )^{9}

\because 1+w+w^{2}=0

\Rightarrow \left ( -w^{2} \right )^{9}=-w^{18}=-1


Option 1)

0

Option 2)

1

Option 3)

\left ( -1+2i \right )^{9}

 

Option 4)

-1

Preparation Products

Knockout JEE Main April 2021

An exhaustive E-learning program for the complete preparation of JEE Main..

₹ 22999/- ₹ 14999/-
Buy Now
Knockout JEE Main April 2022

An exhaustive E-learning program for the complete preparation of JEE Main..

₹ 34999/- ₹ 24999/-
Buy Now
Test Series JEE Main April 2021

Take chapter-wise, subject-wise and Complete syllabus mock tests and get in depth analysis of your test..

₹ 6999/- ₹ 4999/-
Buy Now
JEE Main Rank Booster 2021

This course will help student to be better prepared and study in the right direction for JEE Main..

₹ 13999/- ₹ 9999/-
Buy Now
Test Series JEE Main April 2022

Take chapter-wise, subject-wise and Complete syllabus mock tests and get an in-depth analysis of your test..

₹ 6999/-
Buy Now
Boost your Preparation for JEE Main 2021 with Personlized Coaching
 
Exams
Articles
Questions