Let $S=\{a+b \sqrt{2}: a, b \in Z \}, T_1=\left\{(-1+\sqrt{2})^n: n \in N \right\}$ and $T_2=\left\{(1+\sqrt{2})^n: n \in N \right\}$. Then which of the following statements is (are) $TRUE$?

$(A)$ $Z \cup T_1 \cup T_2 \subset S$

$(B)$ $T_1 \cap\left(0, \frac{1}{2024}\right)=\phi$, 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 $i=\sqrt{-1}$

In the expansion of ${\left( {\frac{x}{2} - \frac{3}{{{x^2}}}} \right)^{10}}$, the coefficient of ${x^4}$is

If $p$ and $q$ be positive, then the coefficients of ${x^p}$ and ${x^q}$ in the expansion of ${(1 + x)^{p + q}}$will be

If the coefficient of ${x^7}$ in ${\left( {a{x^2} + \frac{1}{{bx}}} \right)^{11}}$ is equal to the coefficient of ${x^{ - 7}}$ in ${\left( {ax - \frac{1}{{b{x^2}}}} \right)^{11}}$, then $ab =$

If the coefficients of $(r-5)^{th}$ and $(2 r-1)^{th}$ terms in the expansion of $(1+x)^{34}$ are equal, find $r$

The sum of the binomial coefficients of ${\left[ {2\,x\,\, + \,\,\frac{1}{x}} \right]^n}$ is equal to $256$ . The constant term in the expansion is