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Q.

If $1, α_{1}, α_{2}, α_{3} ....... α_{n - 1}$ are the nth roots of unity then $(1 - α_{1}) (1 - α_{2}) (1 - α_{3}) ....... (1 - α_{n - 1})$ equal to

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a

n

b

1

c

$n^{2}$

d

0

answer is C.

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Detailed Solution

$\begin{array}{l} x^{n} - 1 = (x - 1) (x - α_{1}) (x - α_{2}) \dots (x - α_{n - 1}) \\ \frac{x^{n} - 1}{x - 1} = (x - α_{1}) (x - α_{2}) \dots (x - α_{n - 1}) \\ 1 + x + x^{2} + \dots x^{n - 1} = (x - α_{1}) (x - α_{2}) \dots (x - α_{n - 1}) p u t x = 1 (1 - α_{1}) (1 - α_{2}) \dots (1 - α_{n - 1}) = n \end{array}$

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