First slide

Theory of equations

Question

$If roots of an equation x^{n} - 1 = 0 are 1, a_{1}, a_{2}, \dots, a_{n - 1}, then the value of (1 - a_{1}) (1 - a_{2}) (1 - a_{3}) {............(1-a}_{n - 1}) will be$

Moderate

A

$n$
B

$n^{2}$
C

$n^{n}$
D

None of these

Solution

Clearly, $x^{n} - 1 = (x - 1) (x - a_{1}) (x - a_{2}) \dots (x - a_{n - 1})$
$\begin{array}{l} \Rightarrow \frac{x^{n} - 1}{x - 1} = (x - a_{1}) (x - a_{2}) \dots (x - a_{n}) \\ L H S i s t h e s u m m a t i o n o f s e r i e s i n G P \\ \Rightarrow 1 + x + x^{2} + \dots + x^{n - 1} = (x - a_{1}) (x - a_{2}) \dots (x - a_{n - 1}) \end{array}$
$\begin{aligned} \Rightarrow n = (1 - a_{1}) (1 - a_{2}) \dots (1 - a_{n - 1}) \\ [putting x = 1] \end{aligned}$

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