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Question

If 1,a1,a2,...an-1 are the nth roots of unity, then the value of(1-a1)(1-a2)(1-a3)...(1-an-1) is


A

3

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B

12

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C

n

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D

0

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Solution

The correct option is C

n


Explanation for correct option:

Given,

1,a1,a2,...an-1 are the nth roots of unity So,

xn-1=x-1x-a1...x-an-1xn-1x-1=x-a1x-a2...x-an-1xn-1+xn-2+...+x2+x+1=x-a1x-a2...x-an-1

Substitute x=1

1n-1+1n-2+...+12+1+1=1-a11-a2...1-an-1=1+1+1+...+ntimes

⇒ RHS=n

⇒ LHS=1n-1+1n-2+...+12+1+1

⇒ LHS=1+1+...+1+1+1

⇒ LHS=n

⇒ LHS=RHS

Hence option (C) is correct.


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