If $(1+2x+3x^2)^{10} = a_0+a_1x+a_2x^2+\ldots+a_{20}x^{20}$,then $\frac{a_2}{a_1}$ is equal to

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

For $|x| < \frac{1}{\sqrt{2}}$, the coefficient of $x$ in the expansion of $\frac{(1 - 4x)^2 (1 - 2x^2)^{1/2}}{(4 - x)^{3/2}}$ is

If in the expansion of $(1 + x)^m(1 - x)^n$,the coefficients of $x$ and $x^2$ are $3$ and $-6$ respectively,then $m$ is:

$f(x+h)=0$ represents the transformed equation of the equation $f(x)=x^4+2x^3-19x^2-8x+60=0$. If this transformation removes the term containing $x^3$ from $f(x)=0$,then $h=$

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 $a+b+c=$