Question bank/JEE Main/Mathematics

If
$\alpha = e^{\left( \frac{2\pi i}{7} \right)}\ and\ f(x) = A_{0} + \sum_{k = 1}^{20}\mspace{2mu} A_{k}x^{k}$
and the value of
$f(x) + f(\alpha x) + f\left( \alpha^{2}x \right) + \ldots + \ \ \ \ \ \ \ \ \ \ \ \ \ \ f\left( \alpha^{6}x \right)\ is\ k\left( A_{0} + A_{7}x^{7} + \right.\ A_{14}x^{14}$)
then the value of k is

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$\ We\ know\ 1 + a^{r} + a^{2r} + \ldots\ldots.. + a^{6r} = \left\{ \begin{matrix}
7:r = \ multipleof\ 7 \\
0::r \neq \ multipleof\ 7
\end{matrix} \right.\ $
$f(x) = A_{0} + \sum_{k = 1}^{20}\mspace{2mu} A_{k}x^{k}$
$f(\alpha x) = A_{0} + \sum_{k = 1}^{20}\mspace{2mu} A_{k}\alpha^{k}x^{k}$
add
$f(x) + f(\alpha x) + \ldots.. + f\left( \alpha^{6}x \right) = 7A_{0} + A_{1}\left( 1 + \alpha + \ldots\ldots.. + \alpha^{6} \right)x + \ldots\ldots\ldots\ldots + A_{7}(7)x^{7} + \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ldots\ldots.. + A_{14}(7)x^{14} + \ldots\ldots..$
$= 7A_{0} + A_{1}(0) + A_{2}(0) + \ldots\ldots + A_{6}(0) + A_{7}(7)x^{7} + \ldots\ldots + A_{14}(7)x^{14} + \ldots\ldots + A_{20}(0)$
$$\begin{matrix} = 7\left( A_{0} + A_{7}X^{7} + A_{14}X^{14} \right) \\ \Rightarrow k = 7 \end{matrix}$$

Source: Grass-G / M-28 Mathematics Paper FST 16 Jan.docx