If the sum of all the coefficients of $(\alpha x^2 - 2x + 1)^{2019}$ is equal to the sum of all the coefficients of $(x - \alpha y)^{2019}$,then $\alpha = $

The mean of the coefficients of $x, x^2, \ldots, x^7$ in the binomial expansion of $(2+x)^9$ is $...........$.

If $1 + x^4 + x^5 = \sum\limits_{i = 0}^5 a_i (1 + x)^i$ for all $x$ in $\mathbb{R},$ then $a_2$ is

If $a_k$ is the coefficient of $x^k$ in the expansion of $(1+x+x^2)^n$ for $k=0, 1, 2, \ldots, 2n$,then $a_1+2a_2+3a_3+\ldots+2na_{2n}$ is equal to

Let $x \in \mathbb{R}$ be so small that the powers of $x$ beyond two are insignificant and negligibly small. For such $x$,if $(1-x)^3(2+x)^6$ is approximated by $a+bx+cx^2$,then $a+b+c=$

If $f(x) = 1 + nx + \frac{n(n-1)}{2} x^{2} + \frac{n(n-1)(n-2)}{6} x^{3} + \ldots + x^{n}$,then $f^{\prime \prime}(1)$ is equal to