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  • If   <img alt="1,\alpha 1 , \alpha _2 , \alpha _3 .......and \: \: \alpha 8" src="https://learn.careers360.com/latex-image/?1%2C%5Calpha%201%20%2C%20%5Calpha%20_2%20%2C%20%5Calpha%20_3%20

If   1,\alpha 1 , \alpha _2 , \alpha _3 .......and \: \: \alpha 8 are nine, ninth roots of unity (taken in counter-clockwise

sequence) then


|(2- \alpha _1 )(2- \alpha _3)(2- \alpha _5 )( 2- \alpha _7 )|   is equal to

Option: 1

\sqrt {255 }


Option: 2

\sqrt {511}


Option: 3

\sqrt {1023 }


Option: 4

15 


Answers (1)

best_answer

 

nth roots of unity -

z=\left ( 1 \right )^{\frac{1}{n}}\Rightarrow z=\cos \frac{2k\pi }{n}+i\sin \frac{2k\pi }{n}

Where k=0,1,2,......,(n-1)

-

 

 

(x-1) (x- \alpha _1 ) ( x-\alpha 2 )......... ( x-\alpha _8 ) \equiv x^9 -1 \\\\ (2- \alpha _1 ) ( 2-\alpha 2 )......... ( 2-\alpha _8 ) \equiv 2^9 -1

                Now since  2 -\alpha _1\: and\:2 -\alpha _8  are conjugates of each other

                |2- \alpha _1 |=| 2-\alpha _8 |

                similarly

               |2- \alpha _2 |=| 2-\alpha _7 | , |2- \alpha _3|=| 2-\alpha _6 |

                and          |2- \alpha _4|=| 2-\alpha _5 |

\Rightarrow |(2- \alpha _1)(2- \alpha _3)(2- \alpha _5)(2- \alpha _7)| = \sqrt { 2^9-1 } = \sqrt {511 }

Posted by

Gunjita

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