If 1-2-n-1 are the nth roots of unity of xn-1- then the value of 5-5-2 -5-n-1 is equal to

If 1,ω,ω^2,…,ω^n 1 are the n,n^th roots of unity, then x^n 1=(x 1)(x ω)(x ω^2) …(x ω^n 1) Putting x=5, we get 5^n 1=(5 1)(5 ω)(5 ω^2) …(5 ω^n 1) ⇒ (5 ω)(5 ω^2) ...

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Standard XII

Mathematics

nth Root of a Complex Number

If 1,ω,ω2,…,ω...

Question

If $1, ω, ω^{2}, \dots, ω^{n - 1}$ are the $n^{th}$ roots of unity of $x^{n} = 1,$ then the value of $(5 - ω) (5 - ω^{2}) \dots (5 - ω^{n - 1})$ is equal to

1

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\frac{5^{n} + 1}{4}

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4^{n - 1}

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\frac{5^{n} - 1}{4}

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Solution

The correct option is D $\frac{5^{n} - 1}{4}$
If $1, ω, ω^{2}, \dots, ω^{n - 1}$ are the $n, n^{t h}$ roots of unity, then
$x^{n} - 1 = (x - 1) (x - ω) (x - ω^{2}) \dots (x - ω^{n - 1})$

Putting $x = 5,$ we get
$5^{n} - 1 = (5 - 1) (5 - ω) (5 - ω^{2}) \dots (5 - ω^{n - 1})$
$\Rightarrow (5 - ω) (5 - ω^{2}) \dots (5 - ω^{n - 1}) = \frac{5^{n} - 1}{4}$

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nth Root of a Complex Number

Standard XII Mathematics

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If 1,ω,ω2,…,ωn−1 are the nth roots of unity of xn=1, then the value of (5−ω)(5−ω2)…(5−ωn−1) is equal to

The correct option is D 5n−14If 1,ω,ω2,…,ωn−1 are the n,nth roots of unity, then xn−1=(x−1)(x−ω)(x−ω2)…(x−ωn−1) Putting x=5, we get 5n−1=(5−1)(5−ω)(5−ω2)…(5−ωn−1) ⇒(5−ω)(5−ω2)…(5−ωn−1)=5n−14

If $1, ω, ω^{2}, \dots, ω^{n - 1}$ are the $n^{th}$ roots of unity of $x^{n} = 1,$ then the value of $(5 - ω) (5 - ω^{2}) \dots (5 - ω^{n - 1})$ is equal to