SATHEE: Complex Numbers 5 Question 17

Complex Numbers 5 Question 17

18. If $1, a_{1}, a_{2}, \dots, a_{n - 1}$ are the $n$ roots of unity, then show that $(1 - a_{1}) (1 - a_{2}) (1 - a_{3}) \dots (1 - a_{n - 1}) = n$

(1984, 2M)

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Solution:

Since, $1, a_{1}, a_{2}, \dots, a_{n - 1}$ are $n$ th roots of unity.

$\Rightarrow (x^{n} - 1) = (x - 1) (x - a_{1}) (x - a_{2}) \dots (x - a_{n - 1})$

$\Rightarrow \frac{x^{n} - 1}{x - 1} = (x - a_{1}) (x - a_{2}) \dots . (x - a_{n - 1})$

$\Rightarrow x^{n - 1} + x^{n - 2} + \dots . + x^{2} + x + 1$ $= (x - a_{1}) (x - a_{2}) \dots . (x - a_{n - 1})$ $∵ \frac{x^{n} - 1}{x - 1} = x^{n - 1} + x^{n - 2} + \dots + x + 1$

On putting $x = 1$ , we get $1 + 1 + \dots n$ times

$= (1 - a_{1}) (1 - a_{2}) \dots (1 - a_{n - 1})$ $\Rightarrow (1 - a_{1}) (1 - a_{2}) \dots (1 - a_{n - 1}) = n$

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Complex Numbers 5 Question 17

18. If 1,a1,a2,…,an−1 are the n roots of unity, then show that (1−a1)(1−a2)(1−a3)…(1−an−1)=n

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18. If $1, a_{1}, a_{2}, \dots, a_{n - 1}$ are the $n$ roots of unity, then show that $(1 - a_{1}) (1 - a_{2}) (1 - a_{3}) \dots (1 - a_{n - 1}) = n$