If the sum of all the coefficients of $\left(\alpha x^2-2 x+1\right)^{2019}$ is equal to the sum of all the…
If the sum of all the coefficients of $\left(\alpha x^2-2 x+1\right)^{2019}$ is equal to the sum of all the coefficients of $(x-\alpha y)^{2019}$, then $\alpha=$
-1
0
1
2
Solution
The sum of the coefficient of $\left(\alpha x^2-2 x+1\right)^{2019}$ is $(\alpha-1)^{2019}$. (On putting $\chi=1$ )
and similarly the sum of the coefficients of $(x-\alpha y)^{2019}$ is $(1-\alpha)^{2019}$ (on putting $x=y=1$ )
Now according to the question,
$
\begin{aligned}
& (\alpha-1)^{2019}=(1-\alpha)^{2019} \\
& \Rightarrow \alpha-1=1-\alpha \Rightarrow \alpha=1
\end{aligned}
$