The sum of the coefficients in the expansion of $(1 + x - 3x^2)^{2163}$ will be

The coefficient of $x^{100}$ in the expansion of $\sum_{j=0}^{200} (1 + x)^j$ is

The coefficient of $x^{50}$ in the expansion of $(1+x)^{1000} + x(1+x)^{999} + x^2(1+x)^{998} + \ldots + x^{1000}$ is

The mean of the coefficients of $x, x^2, \ldots, x^7$ in the binomial expansion of $(2+x)^9$ 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 $x$ is so small that all terms containing $x^2$ and higher powers of $x$ can be neglected, then the approximate value of $\frac{(1+\frac{2x}{3})^{-4}(4+5x)^{1/2}}{(9+x)^{3/2}}$ when $x=\frac{6}{371}$, is