[JEE Advanced 1984] If $1,a_1,a_2,\dots,a_{n-1}$ are the $n$ roots of unity, show that $(1-a_1)(1-a_2)\cdots(1-a_{n-1})=n$.
If $1,a_1,a_2,\dots,a_{n-1}$ are the $n$ roots of unity, show that $(1-a_1)(1-a_2)\cdots(1-a_{n-1})=n$.
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Answer: Proved $=n$.
$x^n-1=\prod_{k}(x-a_k)$, so $\frac{x^n-1}{x-1}=1+x+\cdots+x^{n-1}=\prod_{k=1}^{n-1}(x-a_k)$. Putting $x=1$ gives $\prod(1-a_k)=n$.
JEE Advanced 1984 · Complex Numbers — verified solution by the Vidaara Team.
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