If 1- - 2 - n-1 are n roots of unity then - 1- - 2- n-1 equals

The correct option is A ( 1 )^n 1 1,α ,α nbsp;^ 2 ...α nbsp;^ n 1 are the nth roots of unity. Thus, they are solutions of the equation: x ...

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Byju's Answer

Standard XII

Mathematics

Relations between Roots and Coefficients : Higher Order Equations

If 1,α ,α 2...

Question

If $1, α, α^{2} . . . . . α^{n - 1}$ are n roots of unity then , $1. α . α^{2} . . . . α^{n - 1}$ equals

{(- 1)}^{n - 1}

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Solution

The correct option is A ${(- 1)}^{n - 1}$
$1, α, α^{2} . . . α^{n - 1}$ are the nth roots of unity. Thus, they are solutions of the equation: $x^{n} - 1 = 0$ .
Thus, product of roots $= {(- 1)}^{n} * (\frac{- 1}{1})$
(i.e. ${(- 1)}^{n} $ constant term / coefficient of $x^{n}$ )
Thus, the product = ${(- 1)}^{n} (\frac{- 1}{1}) = {(- 1)}^{n} * (\frac{- 1}{1}) = {(- 1)}^{n + 1} = {(- 1)}^{2} * {(- 1)}^{n - 1} = 1 * {(- 1)}^{n - 1} = {(- 1)}^{n - 1}$
Hence, (A) is correct.

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If 1,α,α2.....αn−1 are n roots of unity then ,1.α.α2....αn−1 equals

If $1, α, α^{2} . . . . . α^{n - 1}$ are n roots of unity then , $1. α . α^{2} . . . . α^{n - 1}$ equals