If $f(x)=(2011+x)^n$,where $x$ is a real variable and $n$ is a positive integer,then the value of $f(0)+f^{\prime}(0)+\frac{f^{\prime \prime}(0)}{2 !}+\ldots+\frac{f^{(n-1)}(0)}{(n-1) !}$ is

In the expansion of $(1 + x + x^3 + x^4)^{10}$,the coefficient of $x^4$ is

The sum of the coefficients in the expansion of $(1+x+x^2)^n$ is

Let $S=\{a+b \sqrt{2}: a, b \in Z \}$,$T_1=\{(-1+\sqrt{2})^n: n \in N \}$ and $T_2=\{(1+\sqrt{2})^n: n \in N \}$. Then which of the following statements is (are) $TRUE$?
$(A)$ $Z \cup T_1 \cup T_2 \subset S$
$(B)$ $T_1 \cap (0, \frac{1}{2024}) = \phi$,where $\phi$ denotes the empty set
$(C)$ $T_2 \cap (2024, \infty) \neq \phi$
$(D)$ For any given $a, b \in Z$,$\cos(\pi(a+b \sqrt{2})) + i \sin(\pi(a+b \sqrt{2})) \in Z$ if and only if $b=0$,where $i=\sqrt{-1}$

If $(1 + x - 2x^2)^6 = 1 + a_1x + a_2x^2 + .... + a_{12}x^{12}$,then the expression $a_2 + a_4 + a_6 + .... + a_{12}$ has the value

Let $\alpha, \beta, \gamma$ and $\delta$ be the coefficients of $x^7, x^5, x^3$ and $x$ respectively in the expansion of $(x+\sqrt{x^3-1})^5+(x-\sqrt{x^3-1})^5, x>1$. If $u$ and $v$ satisfy the equations $\alpha u+\beta v=18$ and $\gamma u+\delta v=20$,then $u+v$ equals: